Question: Proofs on SVD (a) Let A be a N x N symmetric matrix. Show that N trace(A) = > >n; n=1 where the {n} are

Proofs on SVD

(a) Let A be a N x N symmetric matrix. Show that N trace(A) = > >n; n=1 where the {n} are the eigenvalues of A. (b) Recall the definition of the Frobenius norm of an M X N matrix: M IAF = m=In=1 Show that R IAllF = trace(AT A) = _ where R is the rank of A and the for } are the singular values of A. (c) The operator norm (sometimes called the spectral norm) of an M X N matrix is max | Ax | |2. TERN, | | |2=1 (This matrix norm is so important, it doesn't even require a designation in its notation if somebody says "matrix norm" and doesn't elaborate, this is what they mean.) Show that 1|All = 01, where on is the largest singular value of A. For which a does | Ax|2 = All . | |2 ? (d) Prove that ||A|

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