用户: Solution/ 试卷: 数值代数及其应用

12024 年秋季学期

1.	Answer the following questions briefly. - $A\in {\mathbb{R}}^{m\times n}$, $B\in {\mathbb{R}}^{n\times p}$, $C\in {\mathbb{R}}^{p\times q}$. How to compute $D=ABC$? - $X$, $Y$ are $3\times 2$ matrices, and $X^{\top }X=Y^{\top }Y=I_2$. What is the angle of these two spaces spanned by $X,Y$? - Compare CGS, MGS, Givens, Householder briefly when using them to compute QR factorization. - What symmetry eigenvalue methods can not be applied to the non–symmetry case? - Forgot :)
2.	If $A$ is column full rank, then prove: $$\left(\begin{array}{cc}I& A^{\top }\\ A& 0\end{array}\right)\left(\begin{array}{c}r\\ x\end{array}\right)=\left(\begin{array}{c}b\\ 0\end{array}\right)$$is equivalent to the least square problem of $(A,b)$.
3.	If $A$ is a strictly column dominate matrix, i.e., ${\sum }_{k\ne i}|A_{ki}|<|A_{ii}|$, then - Please prove that in gepp to compute the $LU$ factorization of $A$, row exchanges never happen. - Please give an upper bound of the condition number of $L$; you can choose any norm you like.
5.	Describe and prove the Implicit Q Theorem.
6.	If $A$ is an upper–triangular matrix with $A_{ii}\ne A_{jj}$ for all $i\ne j$, then please propose a pseudo–code to compute the eigenvectors of $A$, i.e., compute $X,\Lambda$ such that $$A=X\Lambda X^{-1}.$$
7.	In Divide & Conquer method to solve the eigenvalue problem of symmetric matrices, w.l.o.g., we assume that $$D+u{u}^{\top }$$- Elements in diag matrix $D$ are different from each other, - $u_i\ne 0$. Please prove why this does not lose any generality.