91Ó°ÊÓ

Show that if \(A \in \mathbf{R}^{m \times n}\) has rank \(p\), then there exists an \(X \in \mathbf{R}^{m \times p}\) and a \(Y \in \mathbf{R}^{n \times p}\) such that \(A=X Y^{T}\), where \(\operatorname{rank}(X)=\operatorname{rank}(Y)=p\)

Show that if \(Q=Q_{1}+i Q_{2}\) is unitary with \(Q_{1}, Q_{2} \in R^{\times n}\), then the \(2 n\)-by- \(2 n\) real matrix $$ z=\left[\begin{array}{ll} Q_{1} & -Q_{2} \\ Q_{2} & Q_{1} \end{array}\right] $$ is orthogonal.

Show that if \(P\) is an orthogonal projection, then \(Q=I-2 P\) is orthogonal.

$$ \text { Show that if } x \in \mathbf{R}^{n} \text {, then } \lim _{p \rightarrow \infty}\|x\|_{p}=\|x\|_{\infty} $$

What are the singular values of an orthogonal projection?

Suppose \(A \in \mathbf{R}^{n \times n}, b \in \mathbf{R}^{n}\) and that \(\phi(x)=\frac{1}{2} x^{T} A x-x^{T} b\). Show that the gradient of \(\phi\) is given by \(\nabla \phi(x)=\frac{1}{2}\left(A^{T}+A\right) x-b\).

Relate the 2-norm condition of \(X \in \mathbf{R}^{m \times n}(m \geq n)\) to the 2 -norm condition of the matrices, $$ B=\left[\begin{array}{cc} I_{m} & X \\ 0 & I_{n} \end{array}\right] \quad \text { and } \quad C=\left[\begin{array}{c} X \\ I_{n} \end{array}\right] $$

Show that any matrix in \(\mathbf{R}^{m \times n}\) is the limit of a sequence of full rank matrices.

Show that a triangular orthogonal matrix is diagonal.

Show that if \(A \in \mathbf{R}^{m \times n}\) has rank \(n\), then \(\left\|A\left(A^{T} A\right)^{-1} A^{T}\right\|_{2}=1\).