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 2 years ago

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

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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...

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

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