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A paper of importance is M. Anal, vol. This simple but significant property will permit us to deduce a great deal of information concerning the nature of the solutions. On the basis of this knowledge, we shall transform Q x into a simpler form which plays a paramount role in the higher theory of matrices and quadratic forms. Now to resume our story! The Solution of Linear Homogeneous Equations. We require the following fundamental result. From our inductive hypothe- 34 Introduction to Matrix Analysis sis it follows that there exists a nontrivial solution of 3 , z2, Xa,.

Characteristic Roots and Vectors. As a polynomial equation in X, it possesses N roots, distinct or not, which are called the characteristic roots or characteristic values.

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If the roots are distinct, we shall occasionally use the term simple, as opposed to multiple. The hybrid word eigenvalue appears with great frequency in the literature, a bilingual compromise between the German word "Eigenwerte" and the English expression given above. Despite its ugliness, it seems to be too firmly entrenched to dislodge. Associated with each distinct characteristic value X, there is a characteristic vector, determined up to a scalar multiple. This characteristic vector may be found via the inductive route sketched in Sec.

Neither of these is particularly attractive for large values of N, since they involve a large number of arithmetic operations. In actuality, there are no easy methods for obtaining the characteristic roots and characteristic vectors of matrices of large dimension. As stated in the Preface, we have deliberately avoided in this volume any references to computational techniques which can be employed to Diagonalization and Canonical Forms 35 determine numerical values for characteristic roots and vectors.

If X is a multiple root, there may or may not be an equal number of associated characteristic vectors if A is an arbitrary square matrix. These matters will be discussed in the second part of the book, devoted to the study of general, not necessarily symmetric, matrices. For the case of symmetric matrices, multiple roots cause a certain amount of inconvenience, but nothing of any moment.

We will show that a real symmetric matrix of order N has N distinct characteristic vectors. A and A' have the same characteristic values. Does the result hold generally? Show that any scalar multiple apart from zero of a characteristic vector is also a characteristic vector. Show that a similar comment is true for the characteristic roots of AB. Does a corresponding relation hold for the characteristic roots of A and An for n - 3, 4,.

Two Fundamental Properties of Symmetric Matrices. Let us now give the simple proofs of the two fundamental results upon which the entire analysis of real symmetric matrices hinges. Although we are interested only in symmetric matrices whose elements are real, we shall insert the word "real" here and there in order to emphasize this fact and prevent any possible confusion. The characteristic roots of a real symmetric matrix are real. Assume the contrary. We obtain this result and further information from the fact that if the equation holds, then the relation 36 is Also valid.

This means that the characteristic vectors of a real symmetric matrix A can always be taken to be real, and we shall consistently do this. The second result is: Theorem 2. Characteristic vectors associated with distinct characteristic roots of a real symmetric matrix A are orthogonal. This result is of basic importance. Its generalization to more general operators is one of the cornerstones of classical analysis.

A characteristic vector cannot be associated with two distinct characteristic values. Show by means of a 2 X 2 matrix, however, that two distinct vectors can be associated with the same characteristic root. Diagonalization and Canonical Forms 37 AT 7. A matrix of this type is, as earlier noted, called a diagonal matrix. As we shall see, this representation plays a fundamental role in the theory of symmetric matrices.

Reduction of Quadratic Forms to Canonical Form. Let us now show that this matrix transformation leads to an important transformation of Q x. So far we have only established this for the case where the X, are all distinct. As we shall see in Chap. Let A have distinct, characteristic roots which are all positive. Prove along the preceding lines that the characteristic roots of Hermitian matrices are real and that characteristic vectors associated with distinct characteristic roots are orthogonal in terms of the notation [x,y] of Sec.

This again is a particular case of a more general result we shall prove in the following chapter. Let A be a real matrix with the property that A' — —A, a skew-symmetric matrix.

Matrix Analysis with Applications

Show that the characteristic roots are either zero or pure imaginary. Let T be an orthogonal matrix. Show that all characteristic roots have absolute value one. Let T be a unitary matrix. Suppose that we attempt to obtain the representation of 5. We do not dwell upon this point since the proof is a bit more complicated than might be suspected; see Sec. The limit matrix must then be an orthogonal matrix, day T. Since lim m - X.

It is an illustration of a quite useful metamathematical principle that results valid for general real symmetric matrices can always be established by first considering matrices with distinct characteristic roots and then passing to the limit. Positive Definite Quadratic Forms and Matrices. In Sec. Let us now extend this to W-dimensional quadratic forms.

If A is a symmetric matrix with distinct characteristic roots, obtain a set of necessary and sufficient conditions that A be positive definite. Diagonalization and Canonical Forms 41 3. Show that we can write A in the form where the Ei are non-negative definite matrices. Let A and B be two symmetric matrices. Bendixson 4. Show that for any complex matrix A, we have the inequalities t Far more sophisticated results are available.

See the papers by A. Brauer in Duke Math. See also W. So far, we have not dismissed the possibility that a characteristic root may have several associated characteristic vectors, not all multiples of a particular characteristic vector. As we shall see, this can happen if A has multiple characteristic roots. For the case of distinct characteristic roots, this cannot occur. This proof is taken from the book by L. Mirsky, L. The term "latent root" for characteristic value is due to Sylvester. For the reason, and full quotation, see N.

Dunford and J. The term "spectrum" for set of characteristic values is due to Hilbert. Throughout the volume we shall use this device of examining the case of distinct characteristic roots before treating the general case. In many cases, we can employ continuity techniques to deduce the general case from the special case, as in Exercise 7, Sec. Apart from the fact that the method must be used with care, since occasionally there is a vast difference between the behaviors of the two types of matrices, we have not emphasized the method because of its occasional dependence upon quite sophisticated analysis.

It is, however, a most powerful technique, and one that is well worth acquiring. In this chapter, we wish to demonstrate that the results obtained in Chap. The proof we will present will afford us excellent motivation for discussing the useful concept of linear dependence and for demonstrating the Gram-Schmidt orthogonalization technique.

Along the way we will have opportunities to discuss some other interesting techniques, and finally, to illustrate the inductive method for dealing with matrices of arbitrary order. Linear Dependence. Let xl, x2,. If a set of scalars, ci, c2,. If no such set of scalars exist, we say that the vectors are linearly independent. Referring to the results concerning linear systems established in Appendix A, we see that this concept is only of interest if k N, are related by a relation of the type given in l. Show that any set of mutually orthogonal nontrivial vectors is linearly independent.

Given any nontrivial vector in AT-dimensional space, we can always find N — 1 vectors which together with the given JV-dimensional vector constitute a linearly independent set. Gram-Schmidt Orthogonalization. We wish to show 44 Reduction of General Symmetric Matrices 45 that we can form suitable linear combinations of these base vectors which will constitute a set of mutually orthogonal vectors. The procedure we follow is inductive.

We begin by defining two new vectors as follows. Let us next set where now the two scalar coefficients an and o Z2 are to be determined by the conditions of orthogonality These conditions are easier to employ if we note that 1 shows that the equation in 5 is equivalent to the relations These equations reduce to the simultaneous equations which we hope determine the unknown coefficients o2i and We can solve for these quantities, using Cramer's rule, provided that the determinant is not equal to zero.

To show that D2 is not equal to zero, we can proceed as follows. The proof that D3 is nonzero is precisely analogous to that given above in the two-dimensional case. We then say they form an orthonormal set. The determinants Dk are called Gramians. With the same definition of an inner product as above, prove that andthusexpressPintermsoftheexression These polynomials are, apart from constant factors, the classical Legendre polynomials. These polynomials are apart from constant factors the classical Hermite polynomials.

Obtain the analogue of the Gram-Schmidt method for complex vectors. All that was required in the foregoing section on orthogonalization was the nonvanishing of the determinants Dfc. The result is not particularly important at this stage, but the method is an important and occasionally useful one. We shall show that this positivity is a simple consequence of the fact that it is nonzero.

All this, as we shall see, can easily be derived from results we shall obtain further on. It is, however, interesting and profitable to see the depth of various results, and to note how far one can go by means of fairly simple reasoning. Let us begin by observing that D is never zero. This is essentially the same proof we used in Sec. The novelty arises in the proof that D is positive.

Consider the family of quadratic forms defined by the relation where X is a scalar parameter ranging over the interval [0,1]. For all X in this interval, it is clear that P X is positive for all nontrivial w,-. Consequently, the determinant of the quadratic form P X is never zero. Since the determinant of P X is continuous in X for 0 k.

We wish to demonstrate that Let us begin with the result which we obtain from the rule for multiplying determinants. In order to show that a general real symmetric matrix can be reduced to diagonal form by means of an orthogonal transformation, we shall proceed inductively, considering the 2 X 2 case first. Let be a symmetric matrix, and let Xi and a;1 be an associated characteristic root and characteristic vector.

Let the other column be designated by a;2. We wish to show that Reduction of General Symmetric Matrices 51 where Xi and X 2 are the two characteristic values of A, which need not be distinct. This, of course, constitutes the difficulty of the general case. We already have a very simple proof for the case where the roots are distinct.

Let us show first that where and are as yet unknown parameters. The significant fact is that the element below the main diagonal is zero. We have upon referring to the equations in 2. Let us begin with , and show that it has the value zero. This follows from the fact that T'2AT2 is a symmetric matrix, for Finally, let us show that must be the other characteristic root of A.

This completes the proof of the two-dimensional case. Not only is this case essential for our induction, but it is valuable because it contains in its treatment precisely the same ingredients we shall use for the general case. N-dimensional Case. Let us proceed inductively. The elements of the main diagonal X, must then be characteristic roots of A. Proceeding as in the two-dimensional case, we form an orthogonal matrix TI whose first column is x1.

Let the other columns be designated as a? The elements of the first column are all zero except for the first which is Xi b. This shows that these quantities are zero and simultaneously that the matrix AN must be symmetric. The result of all this is to establish the fact that there is an orthogonal matrix TI with the property that with A AT a symmetric matrix.

Let us now employ our inductive hypothesis. Let TV be an orthogonal matrix which reduces An to diagonal form. Let A be a real symmetric matrix. What can be said if A is a complex symmetric matrix? The previous result immediately yields Theorem 3. Theorem 3. A necessary and sufficient condition that A be positive definite is that all the characteristic roots of A be positive.

Similarly, we see that a necessary and sufficient condition that A be positive indefinite, or non-negative definite, is that all characteristic roots of A be non-negative. If the characteristic roots of A are distinct, it follows from the orthogonality of the associated characteristic vectors that these vectors are linearly independent. Let us now examine the general case. Is it true that there exist k linearly independent characteristic vectors with Xi as the associated characteristic root? If so, is it true that every characteristic vector associated with Xi is a linear combination of these k vectors?

To answer the question, let us refer to the representation in 7. Since T is orthogonal, its columns are linearly independent. Hence, if Xi is a root of multiplicity k, it possesses k linearly independent characteristic vectors. It remains to show that any other characteristic vector y associated with Xi is a linear combination of these k vectors. Let y be written as a linear combination of the N columns of T. This shows that y is a linear combination of the characteristic vectors associated with Xi, obtained from the columns of T. From 7. This furnishes a proof of the following special case of a famous result of Cayley and Hamilton.

Theorem 4. Every symmetric matrix satisfies its characteristic equation. As we shall see subsequently, this result can be extended to arbitrary square matrices. Use the method of continuity to derive the Cayley-Hamilton theorem for general symmetric matrices from the result for symmetric matrices for simple roots.

Simultaneous Reduction to Diagonal Form. Having seen that we can reduce a real symmetric matrix to diagonal form by means of an orthogonal transformation, it is natural to ask whether or not we can simultaneously reduce two real symmetric matrices to diagonal form. The answer is given by the following result. Theorem 6. A necessary and sufficient condition that there exist an orthogonal matrix T with the property that is that A and B commute.

The proof of the sufficiency is quite simple if either A or B has distinct characteristic roots. Assume that A has distinct characteristic roots. Since the characteristic roots are assumed distinct, any two characteristic vectors associated with the same characteristic root must be proportional. Then from 3 we can only conclude that Let us, however, see whether we can form suitable linear combinations of the xi which will be characteristic vectors of B. We note, to begin with, that the matrix C — c,y is symmetric for the orthonormality of the xi yields k Consider the linear combination 0,0;'.

Performing similar transformations for the characteristic vectors associated with each multiple characteristic root, we obtain the desired matrix T. The necessity of the condition follows from the fact that two matrices of the form always commute if T is orthogonal.

Simultaneous Reduction to Sum of Squares. As was pointed out in the previous section, the simultaneous reduction of two symmetric matrices A and B to diagonal form by means of an orthogonal transformation is possible if and only if A and B commute. For many purposes, however, it is sufficient to reduce A and B simultaneously to diagonal form by means of a nonsingular matrix.

We wish to demonstrate Theorem 6. Given two real symmetric matrices, A and B, with A positive definite, there exists a nonsingular matrix T such that Proof. It follows that is the desired nonsingular matrix. What is the corresponding result in case A is non-negative definite? It is clear that precisely the same techniques as used in establishing Theorem 2 enable one to establish Theorem 7.

Theorem 7. The Original Maximization Problem. We are now in a position to resolve the problem we used to motivate this study of the positivity and negativity of quadratic forms, namely, that of deciding when a stationary point of a function f xi,xi,. We see then that a sufficient condition that c be a local minimum is that Q be positive definite, and a necessary condition that c be a local minimum is that Q be positive indefinite.

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The criterion given in Sec. If Q vanishes identically, higher order terms must be examined. If N is large, it is, however, not a useful method. Subsequently, in Chap. Perturbation Theory—I. We can now discuss a problem of great theoretical and practical interest. Let A be a symmetric matrix possessing the known characteristic values Xi, A2,. What can we say about the characteristic roots and vectors of A -f- tB, where B is a symmetric matrix and e is a "small" quantity? How small t has to be in order to be called so will not be discussed here, since we are interested only in the formal theory.

If A and B commute, both may be reduced to diagonal form by the same orthogonal transformation. For the sake of simplicity, we shall consider only the case where the characteristic roots are distinct. It will follow from this, that the characteristic vectors of A -f e will be close to those of A.

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One way to proceed is the following. Jhe problem thus reduces to finding approximate expressions for the roots of this equation, given the roots of the original equation and the fact that 6 is small. To do this, let T be an orthogonal matrix reducing A to diagonal form. We do not wish to pursue it in any detail since the method we present in the following section is more powerful, and, in addition, can be used to treat perturbation problems arising from more general operators.

Perturbation Theory—II. Write m, yi as an associated pair of characteristic root and characteristic vector of A -f eB and X,, a;4 as a similar set for A. Then we set To determine the unknown coefficients Xi,, Xz,-,. Similarly, the third equation introduces two further unknowns, the scalar X,-2 and the vector xiz. At first sight this appears to invalidate the method. Let us simplify the notation, writing Then we wish to determine under what conditions upon z the equation has a solution, when X,- is a characteristic root of A, and what this solution is.

Let xl, x9,. Consider the case of multiple roots, first for the 2 X 2 case, and then, in general. Hence, the characteristic roots of B are 10"1, 10"'w,. What is the explanation of this phenomenon? Foray the. This is called the spectral decomposition of A. Then the characteristic roots are pure imaginary or zero.

Show that x and y are orthogonal. Reduction of General Symmetric Matrices 65 4. Prove inductively that if A is a real skew-symmetric matrix of even dimension we can find an orthogonal matrix T such that where some of the m may be zero.

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If A is of odd dimension, show that the canonical form is where again some of the M. Show that the determinant of a skew-symmetric matrix of odd dimension is zero. The determinant of a skew-symmetric matrix of even dimension is the square of a polynomial in the elements of the matrix. Proceeding inductively as before, show that every orthogonal matrix A can be reduced to the form Prove that TA. T' is a positive definite matrix whenever T is an orthogonal matrix and A is a diagonal matrix with positive elements down the main diagonal.

Prove Theorem 5 by means of an inductive argument, along the lines of the proof given in Sec. Establish the analogue of the result in Exercise 9 for unitary matrices. Write down a sufficient condition that the function f xitxt,. If we consider the N — 1 -dimensional matrix where all undesignated elements are zero, show that and determine the characteristic vectors. Show that if the z 1 J. Szego, Orthogonal Polynomials, Am.

## Introduction to Matrix Analysis (Classics in Applied Mathematics)

Tauaaky, and J. Todd Let A and B be real symmetric matrices such that A is non -negative definite. Further results and references to earlier results of Autonne, Wintner, and Murnaghan may be found in J. If A, B, and C are symmetric and positive definite, the roots of have negative real parts Parodi , Let A be a square matrix. A matrix A with this property is called symmetrizable. Show that this matrix A possesses the following properties: a All characteristic roots are real.

Then there exists an "orthogonal" matrix T such that TAT'is diagonal. Extend in a similar fashion the concept of Hermitian matrix. Show that a necessary and sufficient condition that a symmetric matrix A of rank k be idempotent is that k of the characteristic roots are equal to one and the remaining N — k are equal to zero. The notion of rank is defined in Appendix A. If A is a symmetric idempotent matrix, then the rank of A is equal to the A trace of A. Kolmogoroff, Math. The only nonsingular symmetric idempotent matrix is the identity matrix.

C3 and C4 imply Cl and C2. For a proof of the foregoing and some applications, see F. Graybill and G. Marsaglia,1 D. Show that the product of two Lorentz matrices is again a Lorentz matrix. Show that where 0 Reduction of General Symmetric Matrices 69 Let H be a non-negative Hermitian matrix. Using this representation, determine the relations, if any, between multiple characteristic roots of A and the characteristic roots of Aff-i. Using these results, show that the rank of a symmetric matrix can be defined as the order, N, minus the number of zero characteristic roots.

See H. Mirsky, Amer. If A is a symmetric matrix with no characteristic root in the interval [0,6], then A — aI A — bl is positive definite Koto's lemma. Can we use this result to obtain estimates for the location of intervals which are free of characteristic roots of A? Show that L. Mordell, Equationes Mathematicae, to appear 1 Cf. Schur, Math. Determine the minimum of 1 — 6, A z — V over this z-region assuming that A is Hermitian.

See G. Foraythe and G. We suppose that the reader has been exposed to the rudiments of the theory of linear systems of equations. For the occasional few who may have missed this or wish a review of some of the basic results, we have collected in Appendix A a statement and proof of the results required for the discussion in this chapter. The reader who wishes may accept on faith the few results needed and at his leisure, at some subsequent time, nil in the proofs. The result in this section is a particular example of a fundamental principle of analysis which states that whenever a quantity is positive, there exists a formula for this quantity which makes this positivity apparent.

Many times, it is not a trivial matter to find formulas of this type, nor to prove that they exist. See the discussion in G. Hardy, J. Littlewood, and G. The concept of a positive definite quadratic form, as a natural extension of the positivity of a scalar, is one of the most powerful and fruitful in all of mathematics. The paper by Ky Fan indicates a few of the many ways this concept can be used in analysis.

In the appendices at the end of the volume, we also indicate some of the ingenious ways in which quadratic forms may be used in various parts of analysis. A discussion of the diagonalization of complex non-Hermitian symmetric matrices may be found in C. Dolph, J. McLaughlin, and I. Pure and Appl. Questions of this nature arise as special cases of more general problems dealing with the theory of characteristic values and functions of SturmLiouville equations with complex coefficients.

For a further discussion of these problems, and additional references, see R. Brown and I. Green, Matrix Mechanics, Erven P. Noordhoff, Ltd. The book by Green contains a discussion of the matrix version of the factorization techniques of Infeld-Hull. For some interesting extensions of the concept of positive definiteness, see M. In a paper devoted to physical applications, M.

Lax, Localized Perturbations, Phys. XXXVI, pp. Jacobson, Bull. With regard to Exercise 24, see also H. Stenzel, tlber die Darstellbarkeit einer Matrix als Product. Finally, for some interesting connections between orthogonality and quadratic forms, see W. Groebner, t ber die Konstruktion von Systemen orthogonaler Polynome in ein- und zwei-dimensionaler Bereich, Monatsh. Larcher, Proc. For an extensive generalization of the results of the two foregoing chapters, see F.

Atkinson, Multiparametric Spectral Theory, Bull. In particular, we were interested in determining whether x,Ax could assume both positive and negative values. We now add the condition that x simultaneously lie on a set of planes, or alternatively, is a point on a given N — k dimensional plane. Constraints of this nature arise very naturally in various algebraic, analytic, and geometric investigations. As a first step in this direction, we shall carry out the generalization of the algebraic method used in Chap. These results, in turn, will be extended in the course of the chapter. Determinantal Criteria for Positive Definiteness.

Although this is a result of theoretical value, it is relatively difficult to verify. For analytic and computational purposes, it is important to derive more usable criteria. The reason why a criterion in terms of characteristic roots is not useful in applications is that the numerical determination of the characteristic roots of a matrix of large dimension is a very difficult matter. A determinant of order N has AM terms in its complete expansion. Since 73 74 Introduction to Matrix Analysis 10!

S 2, X , it is clear that direct methods cannot be applied, even with the most powerful computers at one's disposal. As we have previously indicated, we will not discuss any aspects of the problem here. From this, we concluded that the relations were necessary and sufficient for A to be positive definite.

Let us now continue inductively from this point. It follows, upon applying the result for 2 X 2 matrices given in 3 , that a set of 1 At the rate of one operation per microsecond, 20! Let us now see if we can persuade the third to assume a more tractable appearance. It follows then that the conditions in 6 may be written in the suggestive form Consequently we have all the ingredients of an inductive proof of Theorem 1.

A necessary and sufficient set of conditions that A be positive definite is that the following relations hold: where We leave the details of the complete proof as an exercise for the reader. Representation as Sum of Squares. Pursuing the analysis a bit further, we see that we can state the following result.

This is the general form of the representation given in 2. Constrained Variation and Finsler's Theorem. Let us now consider the problem of determining the positivity or negativity of the quadratic form Q xi,X2,. Without loss of generality, we can suppose that these k equations are independent. Hence, using these equations, we can solve for k of the xt as linear functions of the remaining N — kt substitute these relations in Q and use the criteria obtained in the preceding sections to treat the resulting N — fc -dimensional problem.

Although this method can be carried through successfully, a good deal of determinantal manipulation is involved. We shall consequently pursue a different tack. However, we urge the reader to attempt an investigation along the lines just sketched at least for the case of one constraint, in order to appreciate what follows. Let us begin by demonstrating the following result of Finsler.

In A'-dimensional y space, write y — Sw to represent this transformation. Our proof is complete upon using the result indicated in Exercise 36 at the end of Chap. Hersteiri 78 Introduction to Matrix Analysis 5. In order to obtain simpler equivalents of these inequalities, we employ the following matrix relation.

Consequently, we see that a sufficient condition that Q be positive for all nontrivial Zj satisfying the linear relation in is that the bordered determinants satisfy the relations for k — 1, 2,. It is clear that the conditions for k — 1, 2,. Referring to 5 , we see that if some element of the sequence of bordered determinants is zero, say the kth, then a necessary condition for 80 Introduction to Matrix Analysis positivity is that the relation be valid.

A further discussion of special cases is contained in the exercises following. Does this yield an inductive approach? Prove that a necessary and sufficient condition that the ffi plane be tangent to the surface is that1 4. Bocher, Introduction to Higher Algebra, Chap. Constrained Maxima 81 6. A Minimization Problem. Closely related to the foregoing is the problem of determining the minimum of x,x over all x satisfying the constraints As usual, the vectors and scalars appearing are real. Without loss of generality, we can assume that the vectors a' are linearly independent.

For any x we know that the inequality is valid. Deduce from this result that the quadratic form is negative lefinite whenever A is positive definite. Give an independent proof of this result by showing that the associated quadratic form cannot be positive definite, or even positive indefinite if A is positive definite. General Values of k.

Let us now return to the problem posed in Sec. It is not difficult to utilize the same techniques in the treatment of the general problem. Since, however, the notation becomes quite cumbersome, it is worthwhile at this point to introduce the concept of a rectangular matrix and to show how this new notation greatly facilitates the handling of the general problem. Once having introduced rectangular matrices, which it would be well to distinguish by a name such as "array," as Cayley himself wished to do, we are led to discuss matrices whose elements are themselves matrices.

We have hinted at this in some of our previous notation, and we shall meet it again in the study of Kronecker products. Rectangular Arrays. We will call this array an M X N matrix. Observe the order of M and N. There is no difficulty in adding two M X N matrices, but the concept of multiplication is perhaps not so clear. This arises as before Constrained Maxima from the iteration of linear transformations.

In many cases, these rectangular matrices can be used to unify a presentation in the sense that the distinction between vectors and matrices disappears. However, since there is a great conceptual difference between vectors and square matrices, we feel that particularly in analysis this blending is not always desirable. Consequently, rectangular matrices, although of great significance in many domains of mathematics, will play a small role in the parts of matrix theory of interest to us here.

Hence, although in the following section we wish to give an example of the simplification that occasionally ensues when this notation is used, we urge the reader to proceed with caution. Since the underlying mathematical ideas are the important quantities, no notation should be adhered to slavishly.

It is all a question of who is master. Let x and y be W-dimensional column vectors. Show that 2. If x is an JV-dimensional column vector, then 9. Composite Matrices. We have previously defined matrices whose elements were complex quantities. Let us now define matrices whose elements are complex matrices, 84 Introduction to Matrix Analysis This notation will be particularly useful if it turns out that, for example, a 4 X 4 matrix may be written in the form with the Aij defined by or, alternatively, with the definitions For various purposes, as we shall see, this new notation possesses certain advantages.

Naturally, we wish to preserve the customary rules for addition and multiplication. Furthermore, the associativity of multiplication is also preserved. Constrained Maxima 85 2. Show that if Af is Hermitian, then Af j is symmetric. The Result for General k. Let us now discuss the general problem posed in Sec. As Theorem 3 asserts, the quadratic form must be positive definite for large X. In order to simplify this relation, we use the same device as employed in Sec. It follows as before that sufficient conditions for positivity are readily obtained.

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The necessary conditions require a more detailed discussion because of the possible occurrence of zero values, both on the part of the coefficients by and the bordered determinants. Let y QijXiXj be a positive definite form. Generalize this result. Use this correspondence and de Moivre's theorem to determine the form of Z" for n - 1, 2,. From Exercise 3, we see that we have the correspondence Using this in the matrix we are led to the supermatrix Constrained Maxima 87 or.

Hence, determine Q. Show that Hubert's inequality Establish Theorem 1 inductively. See C. Determine the minimum value over all z IS. This result plays an important role in prediction theory. See U. Grenander and G. Determine the minimum value over all x,- of the expression where X,- is a sequence of real numbers, Xo C. Seelye, Am. How many stationary values are there, and are they all real? See W. Burnside and A. Panton, Theory of Equations, vol. II, p. A remarkable extension of these inequalities is contained in I. Schur, tlber endliche Gruppen und Hermitesche Formen, Math.

Some interesting results concerning positivity may also be found in I. The question treated here is part of a group of investigations, reference to which may be found in L. The particular result we employ is due to Finsler; cf. Finsler, tlber das Vorkommen definiter und semideh'niter Formen in Scharen quadratischen Formen, Commentarii Mathematicii Helveticii, vol. The result was derived independently by Herstein, using the argument given in the exercise at the end of the section.

We follow his derivation of the conditions here. A number of other derivations appear in later literature, e. Cambridge Phil. The exercises at the end of this section, 3 to 6, illustrate an important heuristic concept which can be made precise by means of the theory of invariants: Whenever we obtain a positivity condition in the form of an inequality, there is a more precise result expressing distances, areas, volumes, etc. In this chapter, we wish to concentrate on the concept of a function of a matrix. We shall first discuss two important matrix functions, the inverse function, defined generally, and the square root, of particular interest hi connection with positive definite matrices.

Following this, we shall consider the most important scalar functions of a matrix, the coefficients in the characteristic polynomial. The problem of defining matrix functions of general matrices is a rather more difficult problem than it might seem at first glance, and we shall in consequence postpone any detailed discussion of various methods that have been proposed until a later chapter, Chap.

For the case of symmetric matrices, the existence of the diagonal canonical form removes most of the difficulty. This diagonal representation, established in Chap. As an example of this we shall prove an interesting result due to Schur concerning the composition of two positive definite matrices. Finally, we shall derive an important relation between the determinant of a positive definite matrix and the associated quadratic form which will be made a cornerstone of a subsequent chapter devoted to inequalities, and obtain also an analogous result for Hermitian matrices.

Functions of Symmetric Matrices. As we have seen, every real symmetric matrix can be represented in the form 90 Functions of Matrices where T is an orthogonal matrix. To obtain an approximate solution to these equations, we determine y as the minimum of the quadratic function x — Ay, x — Ay. If S is a real skew-symmetric matrix, then 7 -f S is nonsingular. If S is a real skew symmetric, then is orthogonal the Cayley transform. Using this result, show that every orthogonal matrix A can be written in the Functions of Matrices 93 form where J is as above.

Show that A"1 can be defined by the relation if A is positive definite. If A is a matrix whose elements are complex integers, i. Show that x,Ax - x,Atx where A. If A and B are alike except for one column, show how to deduce the elements of A"1 from those of B"1. Clement From this deduce the form of A"1. Square Roots. Since a positive definite matrix represents a natural generalization of a positive number, it is interesting to inquire whether or not a positive definite matrix possesses a positive definite square root.

There still remains the question of uniqueness. To settle it, we can proceed as follows. Since B is symmetric, it possesses a representation where S is orthogonal. Hence, both can be reduced to diagonal form by the same orthogonal matrix T. How many symmetric square roots does an N X N positive definite matrix possess? Can a symmetric matrix possess nonsymmetric square roots? If eA — B, B positive definite, and A 6. Parametric Representation. The representation Functions of Matrices 95 furnishes us a parametric representation for the elements of a symmetric matrix in terms of the elements tij of an orthogonal matrix T, namely, which can be used as we shall see in a moment to derive certain properties of the ctij.

Obtain a parametric representation in terms of cos 8 and sin 6 for the elements of a general 2 X 2 symmetric matrix. A Result of I. Using this representation, we can readily Theorem 1. This establishes the required positive definite character. Use Exercise 2 of the miscellaneous exercises of Chap. The Fundamental Scalar Functions. Let us now consider some properties of the scalar functions of A determined by the characteristic polynomial, 96 Introduction to Matrix Analysis In this section the matrices which occur are general square matrices, not necessarily symmetric.

It is called the trace of A, and usually written tr A. It follows from 3 that and a little calculation shows that for any two matrices A and B. This last result is a special case of Theorem 2. If A is nonsingular, we have the relations which yield the desired result. Functions of Matrices 97 In the exercises immediately below, we indicate another approach which does not use any nonalgebraic results.

Although the indefinite integral cannot be evaluated in finite terms, it turns out that the definite integral has a quite simple value. First Proof. Let us now give a second proof. The Jacobian is again equal to 1, and the transformation is one-to-one. Evaluate integrals of the form m, n positive integers or zero, in the following fashion. Write Functions of Matrices 99 and thus How does one treat the general case where either m or n may be odd? If A and B are positive definite, evaluate 4. Evaluate An Analogue for Hermitian Matrices.

By analogy with 9. Then Since the integral is absolutely convergent, it may be evaluated by integration first with respect to x and then with respect to y. Using the relation we see that Hence, Show that and hence, obtain relations connecting k A and tr Ak. Let A, B,. The characteristic roots 01, aj,.

Show that every nonsingular matrix may be represented as a product of two not necessarily real symmetric matrices in an infinite number of ways Voss. For every A there is a triangular matrix such that TA is unitary E. Let P, Q, R, X be matrices of the second order. If A, B, C are positive definite, then the roots of. The characteristic vectors of A are characteristic vectors of p A , for any polynomial p, but not necessarily conversely. Let AB — 0, and let p A,B be a polynomial having no constant term.

Let A and B be positive definite. Show that Introduction to Matrix Analysis Let Y be a positive definite 2 X 2 matrix and X a 2 X 2 symmetric matrix. In other words, this is the region where X is positive definite. See the discussion of Sec. Consequently, if A is symmetric and the leading first minor vanishes, the determinant and its leading second minor have opposite signs. Let X and A be 2 X 2 matrices. Show that Functions of Matrices Weyl2 and in A. Similarly, in the general case, take for the A Introduction to Matrix Analysis Evaluate the matrix function under various assumptions concerning A and the contour C.

Poincari, Sur lea groupes continus, Trans. If A i is nonsingular, then For a generalization, see D. Robinson, Mathematics Magazine, vol. Functions of Matrices 00 An earlier result is due to Schoenberg, Duke Math. J,, If A has the form how does one determine the a where and IR is the identity matrix of dimension n. X is called a reflexive generalized inverse for A if and only if. Gelfand and B. A has the form PD with P a permutation matrix and D a diagonal matrix with positive elements if and only if A and A'1 both have non-negative entries Spira. There is an extensive discussion of functions of matrices in the book by MacDuffee, C.

More recent papers are R. Thomas S. Shores is Professor Emeritus of Mathematics at the University of Nebraska—Lincoln, where he has received awards for his teaching. His research touches on group theory, commutative algebra, mathematical modeling, numerical analysis, and inverse theory. JavaScript is currently disabled, this site works much better if you enable JavaScript in your browser. Mathematics Algebra. Undergraduate Texts in Mathematics Free Preview.