2023秋 线性代数期末试题(回忆版)

Question 1: Not Found

2 Fill in the blanks

2-(1)

Let $u$ be a $n$-dimensional nonzero real column vector and $A=I+u{u}^{\mathrm{\top }}$, where $I$ is the $n\times n$ identity matrix.Then all the distinct eigenvalues of $A$ are $\underset{―}{}$ with algebraic multiplicities

2-(2)

The singular values of the matrix $A=\left[\begin{array}{cc}1& -2\\ -1& 1\\ 2& 1\end{array}\right]$ are $\underset{―}{}$

2-(3)

Let $A=\left[\begin{array}{ccccc}4& -2& 0& 0& 0\\ 0& 0& 1& 1& 0\\ 0& 0& 0& 1& 1\\ 0& 0& 0& 0& 1\end{array}\right]$ .Then $A^{2024}=\underset{―}{}$

2-(4)

Let $x_n$ be a sequence defined by $x_0=0,x_1=1$ and $x_n=2x_{n-1}+x_{n-2}$ for all $n⩾2$ . Then $x_{100}=$ $\underset{―}{}$

2-(5)

Consider the following system of linear equations: $\{\begin{array}{l}{x}_1+2x_2-2x_3=0\\ 2x_1-x_2+a{x}_3=0\\ 3x_1+x_2-x_3=0\end{array}$

If the columns of a nonzero $3\times 3$ matrix $B$ are solutions to the above system, then $a=\underset{―}{}$,|B|= $\underset{―}{}$

3

Consider the $5\times 5$ matrix

$$A=\left[\begin{array}{ccccc}3& 5& 8& 1& -6\\ -3& -4& -7& 1& 9\\ 3& 4& 7& -1& -9\\ -3& -4& -7& 1& 9\\ 2& 3& 5& 0& 5\end{array}\right]$$

The characteristic polynomial of $A$ is $p(x)=-x^3(x-1{)}^2$ .And the reduced row echelon from of A-I is given by

$$A-I\to \left[\begin{array}{lllll}1& 0& 0& -1& -3\\ 0& 1& 0& -1& 0\\ 0& 0& 1& 1& 0\\ 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0\end{array}\right]$$

3-(a)

Find an invertible matrix $S$ such that $S^{-1}AS$ is diagonal

3-(b)

Compute $A^k$ for any positive integer $k$

4

Find the matrix $Q$ in the $QR$-factorization of the matrix $A=\left[\begin{array}{lll}1& 1& 0\\ 1& 0& 1\\ 0& 1& 1\end{array}\right]$

5

Let $A=\left[\begin{array}{ccc}0& -i& 0\\ i& 1& i\\ 0& -i& 0\end{array}\right]$

5-(a)

Show that $A$ is Hermitian

5-(b)

Find the eigenvalues and eigenvectors of $A$

5-(c)

Can we find a unitary matrix $U$ such that $U^{H}AU$ is diagonal？If yes, find one

such matrix. Otherwise, explain

6

Let $B$ be an $n\times n$ real symmetric matrix. A quadratic form $g(y)=y^{\mathrm{\top }}$ By is called negative define definite if $y^{\mathrm{\top }}By<0$ for all nonzero real vectors $y\in R^n$. Consider the following quadratic form

$$f(x_1,x_2,x_3)=(\lambda -3)x_1^2+4x_1,x_2+\lambda x_2^2+(\lambda -1)x_3^2$$

6-(a)

Find a real symmetric matrix $A$, such that $f(x_1,x_2,x_3)=x^{\mathrm{\top }}Ax$, where $x=\left[\begin{array}{l}{x}_1\\ x_2\\ x_3\end{array}\right]$

6-(b)

Determine the range of $\lambda$ such that $f(x_1,x_2,x_3)$ is negative definite

7

Let $A,B$ be two $n\times n$ real matrices. Suppose $A$ has $n$ distinct eigenvalues and $AB=BA$

7-(a)

Show that $B$ is diagonalizable

7-(b)

Suppose $n=3$. Show that there exists a polynomial of degree at most $2,g(x)$, such that $B=f(A)$

7-(c)

Suppose $n>3$. Show that there exists a polynomial of degree at most $n-1,f(x)$, such that $B=f(A)$