P.0.2 Show that (a) the diagonal entries of a Hermitian matrix are real; (b) the diagonal...

P.0.2 Show that (a) the diagonal entries of a Hermitian matrix are real; (b) the diagonal entries of a skew-Hermitian matrix purely imaginary; c) the diagonal entries of a skew-symmetric matrix are zero.

P.0.5 Let A ∈ Mn be invertible. Use mathematical induction to prove that (A-1)k = (Ak)-1 for all integers k.

P.0.25 Let A ∈ Mn be idempotent. Show that A is invertible if and only if A = I

P.0.26 Let A,B ∈ Mn be idempotent. Show that tr((A-B)3) = tr(A-B).

Expert Solution

Colby Messinger answered 1 year ago

(a) Show that the diagonal entries of a positive definite matrix are positive numbers. (b) Show...

(a) Show that the diagonal entries of a positive definite matrix are positive numbers. (b) Show that if B is a nonsingular square matrix, then BTB is an SPD matrix.(Hint. you simply need to show the positive definiteness, which does requires the nonsingularity of B.)

For the matrix A, find (if possible) a nonsingular matrix P such that P−1AP is diagonal....

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Diagonalize the matrix (That is, find a diagonal matrix D and an invertible matrix P such...

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(2) A matrix A is given. Find, if possible, an invertible matrix P and a diagonal...

(2) A matrix A is given. Find, if possible, an invertible matrix P and a diagonal matrix D such that P −1AP = D. Otherwise, explain why A is not diagonalizable. (a) A = −3 0 −5 0 2 0 2 0 3 (b) A = 2 0 −1 1 3 −1 2 0 5 (c) A = 1 −1 2 −1 1 2 2 2 2

Show that the set of all n × n real symmetric matrices with zero diagonal entries...

Show that the set of all n × n real symmetric matrices with zero diagonal entries is a subspace of Rn×n

Give an example of a square matrix A such that all the diagonal entries are positive...

Give an example of a square matrix A such that all the diagonal entries are positive but A is not positive definite

Let P ( A ) = 0.5, P ( A ∩ B ) = 0.2 and...

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Orthogonally diagonalize the matrix by finding an orthogonal matrix Q and a diagonal matrix D such...

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An experiment consists of four outcomes (A, B, C, D) with P(A) = 0.2, P(B) =...

An experiment consists of four outcomes (A, B, C, D) with P(A) = 0.2, P(B) = 0.3, and P(C) = 0.4. The P(D) is 0.500 0.024 0.100 0.900 Given that event E has a probability of 0.3, the probability of the complement of event E cannot be determined with the above information can have any value between zero and one must be 0.7 is 0.3 If P(A) = 0.38, P(B) = 0.83, and P(A Ç B) = 0.57; then P(A È...

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