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124-25 学年第二学期习题

Problem 1.	[10 pt] For $x\in {\mathbb{R}}^m$ and $A\in {\mathbb{R}}^{m\times n}$, verify the following inequalities and give examples of a nonzero vector or matrix for which equality is achieved. (a) $\Vert x{\Vert }_{\infty }\le \Vert x{\Vert }_2\le \sqrt{m}\Vert x{\Vert }_{\infty }$ (b) $\frac{1}{\sqrt{n}}\Vert A{\Vert }_{\infty }\le \Vert A{\Vert }_2\le \sqrt{m}\Vert A{\Vert }_{\infty }$ (c) Show equivalence between induced matrix norms $\Vert \cdot {\Vert }_p$ and $\Vert \cdot {\Vert }_q$ for $1\le p<q\le \infty$
Problem 2.	[10 pt] For each of the following statement, find an example matrix. (a) a matrix whose singular values are the same as its eigenvalues; (b) a matrix whose singular values are the same as the absolute values of its eigenvalues (the matrix has at least one negative eigenvalue); (c) a matrix whose singular values are not the same as the absolute values of its eigenvalues;
Theorem I.	Let $A\in {\mathbb{R}}^{m\times n}$ with SVD $\{{\sigma }_i,u_i,v_i{\}}_{i=1}^{\min (m,n)}$. For $0\le r\le \min (m,n)$, the matrix $A_r$ as defined,$$A_r=\sum _{i=1}^r{u}_i{\sigma }_i{v}_i^{\ast }$$satisfies$$\Vert A-A_r{\Vert }_{\mathrm{F}}=\underset{\begin{array}{c}B\in {\mathbb{C}}^{m\times n}\\ \mathrm{rank}(B)\le r\end{array}}{\inf }\Vert A-B{\Vert }_{\mathrm{F}}.$$
Problem 3.	[20 pt] Prove the Theorem I.
Problem 4.	[15 pt] For nonzero projector $P\in {\mathbb{C}}^{m\times m}$, show that $\Vert P{\Vert }_2\ge 1$ with equality iff $P$ is an orthogonal projector.
Problem 5.	[20 pt] Implement a matrix-matrix multiplication algorithm in C++.
Problem 6.	[10 pt] Call “dgemm” function in LAPACK for a matrix-matrix multiplication in C++.
Problem 7.	[10 pt] Apply your code in Problem 5 and Problem 6 to two random matrices of size 5000×5000 and compare their runtime.