Normal matrices have A A = AA . For real matrices, A A = AA includes symmetric

Chapter 6, Problem 29

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Normal matrices have A A = AA . For real matrices, A A = AA includes symmetric, skew-symmetric, and orthogonal. Those have real A, imaginary A, and IAI = 1. Other normal matrices can have any complex eigenvalues A. Key point: Normal matrices have n orthonormal eigenvectors. Those vectors Xi probably will have complex components. In that complex case orthogonality means x T x j = 0 as Chapter 10 explains. Inner products (dot products) become x T y. The test/or n orthonormal columns in Q becomes Q T Q = 1 instead 0/ QT Q = 1. A has 11 orthonormal eigenvectors (A = Q A Q T) if and only if A is normal. -T -T -T-T (a) Start from A = QA Q with Q Q = 1. Show that A A = AA : A is normal. -T -T -T (b) Now start from A A = A A . Schur found A = Q T Q for every matrix A, with a triangular T. For normal matrices we must show (in 3 steps) that this T will actually be diagonal. Then T = A. -T -T -T -T -T Step 1. Put A = Q T Q into A A = AA to find T T = T T . [ a b ] -T -T Step 2. Suppose T = 0 d has T T = TT . Prove that b = O. Step 3. Extend Step 2 to size n. A normal triangular T must be diagonal