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9.3. Infinite Series Of Matrices. Norms Of Matrices Let A aij be an n n matrix of scalar entries. Given a scalar function f x , we can construct a matrix- valued function f A by replacing x with A in f x . Hopefully, useful characteristics of f x would be carried over to f A . For example, we expect the exponential e A to possess the laws of exponents, namely, t s A for all scalars s, t. (9.8) etAesA e O e I (9.9) where O and I are the n n zero and identity matrices, respectively. However, for transcendental functions, the substitution scheme does not tell us how to calculate the numerical value of f A for a given A. Here, as in the scalar case, the problem is solved by defining the function in terms of a (convergent) power series. 1 k x k 0 k ! ex For example, 1 k A k 0 k ! eA The next question naturally concerns the convergence of power series of matrices. Definition: Convergent Series Of Matrices Given an infinite sequence of m n matrices C c , the series of matrices k ij k C k 1 is convergent iff all mn series k c k 1 for i 1, k ij are convergent. , m and j 1, (9.10) In which case, we write A Ck aij aij cij with k k 1 k 1 Definition: ,n Norm Of A Matrix m n Let M be the space of all m n matrices of scalar entries. The norm is a mapping M mn F by A A where F is the scalar field R or C, such that the following axioms are satisfied (a) A 0 with A 0 iff A O [Positivity] (b) cA c A cF [Homogeneity] (c) A B A B [Triangle inequality] Comment As in the case of vectors, there are many realization of the norm. Let A aij be an m n matrix of scalar entries. m We shall define n A aij (9.11) i 1 j 1 Theorem 9.1. Definition (9.11) is indeed a norm. Furthermore, AB A B . Proof We shall only prove the inequality AB A B , which can be viewed as a generalized form of the Cauchy- Schwarz inequality. Assuming A aij is m n and B bij is n p, we have p m AB i 1 j 1 p m m n a k 1 n n ik bkj p n aik bkj i 1 j 1 k 1 aik blj m p n n aik blj kl i 1 j 1 k 1 l 1 A B i 1 j 1 k 1 l 1 Comment In the special case where A B are n n square matrices, the C-S inequality becomes A2 A 2 which is easily generalized to give Ak A k for all k 1,2,3, Theorem 9.2. Let Test For Convergence Of Matrix Series C c be an infinite sequence of n n matrices such that the series of norms k ij k C k 1 k converges. Then the matrix series C k 1 k also converges. Proof Obviously, cij k n c l ,m 1 k lm Ck Therefore, the convergence of the series k 1 Ck implies the convergence of k 1 for all i and j. This in turn implies the convergence of c k 1 Definition (9.10) then proves the theorem. c k ij for all i and j. k ij