Let A S n + be a symmetric, positive semi-definite matrix. 1. Show that the trace,
Question:
Let A ∈ Sn+ be a symmetric, positive semi-definite matrix.
1. Show that the trace, trace A, and the Frobenius norm,ΙΙAΙΙF, de-pend only its eigenvalues, and express both in terms of the vector of eigenvalues.
2. Show that
3. Identify classes of matrices for which the corresponding lower bound on the rank is attained.
Transcribed Image Text:
(trace A)² ≤ 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 the full answer
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Related Book For
Optimization Models
ISBN: 9781107050877
1st Edition
Authors: Giuseppe C. Calafiore, Laurent El Ghaoui
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