Question: Let A S n + be a symmetric, positive semi-definite matrix. 1. Show that the trace, trace A, and the Frobenius norm,A F, de-pend

Let A ∈ Sⁿ₊be a symmetric, positive semi-definite matrix.

1. Show that the trace, trace A, and the Frobenius norm,ΙΙAΙΙ_F,de-pend only its eigenvalues, and express both in terms of the vector of eigenvalues.

2. Show that

(trace A) rank(A)||A||} .

3. Identify classes of matrices for which the corresponding lower bound on the rank is attained.

(trace A) rank(A)||A||} .

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1 2 The rank of A is simply the number of eigenvalues that are nonzero Without loss of generality we assume that the eigenvalues are ordered in decreasing fashion The inequality follows from the CauchySchwartz inequality with 1 r the vector of ones in R r we have which proves the desired result 3 We begin with an eigenvalue ... View full answer

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