(a) If A = A" and P is the orthogonal projection onto the column space of A, then AP = PA. (b) If A = A" and all the eigenvalues of A are positive, then there is a matrix B such that Bª = A. (c) If A is invertible and o is a singular value of A then 1/o is a singular value of A-!. (d) If A and B are similar square matrices, then every singular value of A is also a singular value of B. (e) If A is real symmetric matrix then A is similar to a real diagonal matrix.
(a) If A = A" and P is the orthogonal projection onto the column space of A, then AP = PA. (b) If A = A" and all the eigenvalues of A are positive, then there is a matrix B such that Bª = A. (c) If A is invertible and o is a singular value of A then 1/o is a singular value of A-!. (d) If A and B are similar square matrices, then every singular value of A is also a singular value of B. (e) If A is real symmetric matrix then A is similar to a real diagonal matrix.
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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Transcribed Image Text:(a) If A = A" and P is the orthogonal projection onto the column space of A, then AP = PA.
(b) If A = A" and all the eigenvalues of A are positive, then there is a matrix B such that Bª = A.
(c) If A is invertible and o is a singular value of A then 1/o is a singular value of A-!.
(d) If A and B are similar square matrices, then every singular value of A is also a singular value
of B.
(e) If A is real symmetric matrix then A is similar to a real diagonal matrix.
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