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75) is equivalent to solving the n systems Ak, = q, to obtain H. Let A be an n x n matrix. For some nonzero column vector v = col ( v l , . . v,) it may happen that, for some scalar A , Av = hv. 78) If this occurs, we say that h is an eigenvalue of A and that v is an eigenvector of A, associated with the eigenvalue A . The concept of eigenvalue has important applications in many branches of physics. An important example is the spectrum-f light, of an atom, of a nucleus. The frequencies occurring in the spectrum correspond to the eigenvalues of a matrix (or of a suitable generalization of a matrix).
A , = a l l + . . + a , , , , . T h e n u n l b e r a l ~ + . . +a,,, iscalledthe trace of A. d) Prove from the rcsult of (c) that A and B havc equal traces. [ I ] is not similar to a diagonal matrix. ] 7. Prove that the matrix A = Chapter 1 Vectors and Matrices 8. Prove the following: a) Every square matrix is similar to itself. b) If A is similar to B and B is similar to C, then A is similar to C. 12 THETRANSPOSE Let A = (a;,) be an m x n matrix. We denote by A' the n x m matrix B = ( b i j )such that hi, = a,; for i = 1 , .
Prove the following: + + a) If z l and 12 are complex numbers, then ( z I z 2 ) = ? I 2 2 and zlzz = 7 172. -b) -If A and B are matrices and c is a complex scalar, then ( A B ) = A B , AB = A B , ( c A ) = FA. C) Every complex matrix A can be written uniquely as A 1 i A 2 ,where A l and A? arc real matrices, and A I = + ( A+A),A? = ( 2 i ) - ' ( ~ A ) . We call A l the real part of A. A2 the imaginary pariof A. + + + - d ) If A is a square matrix, then (A)'= (A') and, if A is nonsingular, then ( A p ' )= ( A ) - ' .