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Prove the next arguments about normal matrix A € Cnx" and all of its eigenvalues EC 1. A*=A^-1 (Unitary A) if and only if |A| = 1 2. A*=A (Symmetrical A) if and only if X E R 3. (Ax, x) ≥ 0 for every EC" (Positive A) if and only if > > 0 2

Prove the next arguments about normal matrix A € Cnx" and all of its eigenvalues EC 1. A*=A^-1 (Unitary A) if and only if |A| = 1 2. A*=A (Symmetrical A) if and only if X E R 3. (Ax, x) ≥ 0 for every EC" (Positive A) if and only if > > 0 2

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Prove the next arguments about normal matrix A E C"*" and all of its eigenvalues A E C: 1.A=A^-1 (Unitary A) if and only if |A| = 1 2. A*=A (Symmetrical A) if and only if A E R 3 .(Aa, r) 20 for every r E C" (Positive A) if and only if A > 0 4. A* = A = A² (Orthogonal projection A) if and only if A E {0, 1}
Transcribed Image Text:Prove the next arguments about normal matrix A E C"*" and all of its eigenvalues A E C: 1.A=A^-1 (Unitary A) if and only if |A| = 1 2. A*=A (Symmetrical A) if and only if A E R 3 .(Aa, r) 20 for every r E C" (Positive A) if and only if A > 0 4. A* = A = A² (Orthogonal projection A) if and only if A E {0, 1}
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