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**Extra info for Linear Algebra III**

**Example text**

Show (A∗ ) = A and (AB) = B ∗ A∗ . 2. 9. 3. Show that if A is an n × n matrix which has an inverse then A+ = A−1 . 4. Using the singular value decomposition, show that for any square matrix A, it follows that A∗ A is unitarily similar to AA∗ . 5. Let A, B be a m × n matrices. Deﬁne an inner product on the set of m × n matrices by (A, B)F ≡ trace (AB ∗ ) . Show this is an inner product ∑nsatisfying all the inner product axioms. Recall for M an n × n matrix, trace (M ) ≡ i=1 Mii . The resulting norm, ||·||F is called the Frobenius norm and it can be used to measure the distance between two matrices.

Recall the following symbols and their meaning. lim sup an , lim inf an n→∞ n→∞ They are respectively the largest and smallest limit points of the sequence {an } where ±∞ is allowed in the case where the sequence is unbounded. They are also deﬁned as lim sup an ≡ n→∞ lim inf an ≡ n→∞ n→∞ n→∞ lim (sup {ak : k ≥ n}) , lim (inf {ak : k ≥ n}) . Thus, the limit of the sequence exists if and only if these are both equal to the same real number. 2 Let J be a p × p Jordan matrix J1 .. J = . Js where each Jk is of the form Jk = λk I + Nk in which Nk is a nilpotent matrix having zeros down the main diagonal and ones down the super diagonal.

Let y = R∗ Rx − x to conclude that for all x, R∗ Rx − x = 0 which says R∗ R = I since x is arbitrary. The decomposition in the following is called the right polar decomposition. 2 Let X be a Hilbert space of dimension n and let Y be a Hilbert space of dimension m ≥ n and let F ∈ L (X, Y ). Then there exists R ∈ L (X, Y ) and U ∈ L (X, X) such that F = RU, U = U ∗ , (U is Hermitian), all eigenvalues of U are non negative, U 2 = F ∗ F, R∗ R = I, and |Rx| = |x| . 3, there is an orthonormal basis of eigenvectors, {v1 , · · · , vn } such that F ∗ F v i = λi v i , F ∗ F = n ∑ i=1 λi v i ⊗ v i .