True and False. Please Explain. Let E be an elementary matrix and A an n times n

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True and False. Please Explain.

Let E be an elementary matrix and A an n times n matrix. Then EA and A have the same eigenvalues. (vii) T F: Two eigenvectors of a symmetric matrix A corresponding to two distinct eigenvalues are orthogonal to each other. (viii) T F: If a square matrix A is diagonalizable with eigenvalues lambda_1, lambda_2, ..., lambda_n, then there is a unique matrix P such that P^-1 AP = diag{lambda_1, lambda_2, ... lambda_n}, where diag{lambda_1, lambda_2, ..., lambda_n} is the diagonal matrix with the eigenvalues on the main diagonal. (ix) T F: If a square matrix A is diagonalizable. then so is A^T. (x) T F: If lambda is an eigenvalue of a square matrix A, then lambda^k must be an eigenvalue of A^k for any positive integer k. (xi) T F: Let A be an n times n real matrix. Then A and A^T have the same characteristic polynomial. (xii) T F: Let f(t) = a_0 + a_1 t +... + a+n t^n be a polynomial. Then every eigenvector of A is also an eigenvector of f(A) = a_0 I + a_1 A + ... + a_n A^n. (xiii) T F: Let A be an n times n positive definite real symmetric matrix. Then every eigenvalue of A is positive. (xiv) T F: Every real symmetric matrix is diagonalizable. (xv) T F: Every eigenvalue of a real symmetric matrix is real.

Explanation / Answer

(vi). The staement isFalse.If E is an elementary matrix and A a nxn matrix, then EA represents a row operation on A which changes the eigenvalues of the matrix.

(vii) The statement is True. If A is a symmetric matrix, then AA* = A2 so that A is a normal matrix. Then, as per the spectral theorem, the eigenvectors of A corresponding to two distinct eigenvalues are orthogonal to each other.

(viii) The statement is True. It is a direct inference of the process of diagonalization. If the diagonal matrix D = diag{ 1 , 2,….,n } maintains the order of eigenvalues 1 , 2,….,n of A , then the matrix P consisting of the corresponding eigenvectors of A as columns, is also unique. Hoeever, if there is a permutation of the eigenvalues in D, then P will also change.

(ix) The statement is True. A is diagonalizable means A is similar to a diagonal matrix D i.e. there is an invertible matrix P such that A = P-1 DP. Then AT = (P-1 D P)T = PT DT (P-1)T = PT D(PT )-1 Thus AT is similar to D so that AT is diagonalizable .

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