For a matrix A in Exercise A.34, find a generalized inverse a. that is symmetric

Question

For a matrix A in Exercise A.34, find a generalized inverse a. that is symmetric b. that is not symmetric c. that has rank 4 (and hence nonsingular) d. so that A is a generalized inverse of it. e. So that A is not a generalized inverse of it. If P is idempotent, show that (I - P) is also idempotent. If P is symmetric and idempotent, show that the Pythagorean relationship holds ||y||^2 = ||Py||^2 + | (I - P) y||^2 Prove the results about the trace operator in Resale A.12. Show that the trace of the matrix 1, is equal to its rank. Find the trace and determinant for each of the following four matrices [13 4 4 7] [2 0 0 0 2 0 0 0 -1] [12 x 8 12] [9 10 10 30] Show that all eigenvectors of s symmetric matrix with distinct eigenvalues are linearly independent. Prove that for a symmetric matrix |A| = lambda_1 + .... + lambda_ where lambda_ are the eigenvalue of A. Use Result A.18(c) to prove: if A, B are m times n, then |I_ + AB^T| = |I_ + B^T A|. Find the eigenvalues and eigenvectors of the following four matrixes: [13 4 4 7] [2 0 0 0 2 0 0 0 -1] [12 8 8 12] [9 10 10 30] Find the eigenvalues and eigenvectors for a matrix of the following form: [a b b a] Compute the spectral decompositions of the matrix in Example A8.

Explanation / Answer

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Solution for A. 36

Given that P is an Idempotent matrix.

Conisder (I - P)2 = (I - P)(I - P)
= I - P - P + P.P
= I - P - P + P2
= I - P - P + P ( Since P2 = P)
= I - P
Hence I - P is idempotent.

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