2024春 高代下 final(H) (回忆版)

Problem 1

Let $X$ be the set of all symmetric $n\times n$ matrices over $R$ , and $Y\subseteq X$ the subset of all positive semidefinite matrices

1-(i)

For two matrices $A,B\in X$ , define $A⩾B$ if $A-B\in Y$. Prove that $⩾$ is a partial order, ie

• (1) $A⩾A$
• (2) $A⩾B$ and $B⩾A$ implies $A=B$
• (3) $A⩾B$ and $B⩾C$ implies $A⩾C$

1-(ii)

Prove that $Y$ is convex, ie. for $A,B\in Y$ , we have $tA+(1-t)B\in Y$ for all real $t\in [0,1]$

1-(iii)

Let $A,B\in Y$. Suppose that $AB=BA$. Prove that $AB\in Y$

1-(iv)

Let

$$Z:=\{A\in X:\mathrm{tr}(AB)⩾0\text{ for all }B\in Y\},$$

where tr means trace. It is a known fact that $\mathrm{tr}(AB)=\mathrm{tr}(BA)$ for matrices A, B with compatible sizes. Prove that $Z=Y$

Problem 2

2-(i)

Let $X:=(-0.01,2\pi )\times (-0.01,2\pi )$. Consider $f:X\to R$ given by $f(x,y)=(\cos x)(\cos y)$. Find all its local maximals(both points and values) and local minimals(both points and values)

2-(ii)

Let $X$ be the set of real $2\times 2$ orthogonal matrices. Consider $f:X\to R$ given by $A\mapsto \mathrm{tr}{A}^2$ where tr means trace. Find all its go global maximum and minimum(both points and values)

Problem 3

We define an equivalence relation $\sim$ on $M_n(k)$ by

$$A\sim B\text{ if }B=S^{\mathrm{\top }}AS\text{ for some invertible }S\in M_n(k)\text{. }$$

3-(i)

For $k=C$ and $n=2$ , find a complete set of representatives for $M_n(k)/\sim$

3-(ii)

For $k=R$ and $n=2$ , find a complete set of representatives for $M_n(k)/\sim$

Problem 4

Let $M$ be a finite inner product space over $C$. For an ordered basis $B$ , let $G_B$ be its Gram matrix. A pair of dual bases consists of an ordered basis $B=(b_1,\cdots ,b_m)$ of $M$ and another ordered basis $C=(c_1,\cdots ,c_m)$ of $M$ such that

$${⟨b_i,C_j⟩}_M={\delta }_{ij}$$

where ${\delta }_{ij}:=1$ if $i=j$ and ${\delta }_{ij}:=0$ if $i\ne j$

4-(i)

Let $b_1,\cdots ,b_m$ and $c_1,\cdots ,c_m$ be dual bases. Give a simple formula for an element $v\in M$ as a linear combination of $b_1,\cdots ,b_m$

4-(ii)

Let $b_1,\cdots ,b_m$ be an ordered basis of $M$. Prove that there exist $c_1,\cdots ,c_m\in M$ such that $b_1,\cdots ,b_m$ and $c_1,\cdots ,c_m$ are dual bases

4-(iii)

Prove that, if $B$ and $C$ are dual bases, then $G_B$ and $G_c$ are inverses to each other

4-(iv)

Disprove that, if $G_B$ and $G_C$ are inverses to each other, then $B$ and $C$ are dual bases

Problem 5

Consider square matrices over C. It is a known fact that every matrix is upper triangularizable. Prove the following statements

5-(i)

Every matrix is unitarily triangularizable

5-(ii)

Eigenspaces with distinct eigenvalues of normal matrix are orthogonal

5-(iii)

Every normal matrix is unitarily diagonalizable

5-(iv)

A matrix is unitarily diagonalizable if and only if it is normal

Problem 6

Let $k:=c$. For a $A\in M_n(k)$, let

$$e^A:=\sum _{i=0}^{\mathrm{\infty }}\frac{A^i}{i!},$$

where we adopt the convention that $A^{\circ }=I$. It is known facts that $e^A$ always exists (the series converges) and $e^{A+B}=e^A+e^B$ for $[A,B]=0$. A logarithm of $A$ is a matrix $B$ such that $e^B=A$

6-(i)

Find all logarithms of the $2\times 2$ identity matrix

6-(ii)

Let $\theta \in R$ such that $\sin \theta \ne 0$. Let $A_{\theta }:=\left[\begin{array}{rr}\cos \theta & -\sin \theta \\ \sin \theta & \cos \theta \end{array}\right]$

Find all logarithms of $A_{\theta }$

6-(iii)

Prove that, for a unitary matrix $A$, there exists a skew -Hermitian $(B^t=-B)$ logarithm B

6-(iv)

Prove that a logarithm of $A$ exist if and only if $A$ is invertible

Problem 7

Let $M$ be a finite $k$-module over an algebraically closed field $k$. Let $g:=\mathrm{End}(M)$. For each $x\in g$, let $x=x_s+x_n$ be its (unique) Jordan-Chevalley decomposition in $g$. For each $x\in g$, let ad $x:g\to g$ be the (a prior not necessarily linear) map given by $y\mapsto [x,y]$, where $[x,y]:=xy-yx$

7-(i)

Prove that the map ad $x$ is an element in $\mathrm{End}(g)$, ie. and $x$ is a linear transformation

7-(ii)

Prove that the map ad $x_s$ is semisimple in $\mathrm{End}(g)$

7-(iii)

Prove that the map ad $x_n$ is nilpotent in $\mathrm{End}(g)$

7-(iv)

Prove that $[\mathrm{ad}{x}_s,\mathrm{ad}{x}_n]=0$ in $\mathrm{End}(g)$

Problem 8

Let $k:=C,d⩾1$, and $C_0,\cdots ,C_{d-1}\in k$. Let $p:=x^d+C_{d-1}{x}^{d-1}+\cdots +c_{1}x+C_0$ and suppose that $p={(x-{\lambda }_1)}^d\cdots {(x-{\lambda }_t)}^{dt}$ for distinct ${\lambda }_1,\cdots ,{\lambda }_t\in k$. The companion matrix of $p$ is

$$C:={\left[\begin{array}{ccccc}0& 0& \cdots & 0& -C_0\\ 1& 0& \cdots & 0& -C_1\\ 0& 1& \cdots & 0& -C_2\\ ⋮& ⋮& \ddots & 0& ⋮\\ 0& 0& \cdots & 1& -C_{d-1}\end{array}\right]}_{d\times d}$$

8-(i)

For $d=2$, find an explicit similarity from $C$ to its Jordan normal form

8-(ii)

Consider a recurrence $f_0=0,f_1=1$ and $f_{n+2}=a{f}_{n+1}+b{f}_n$, where $a,b,f_i\in C$. Given an explicit formula (without using matrices) for $f_n$ in terms of $a,b,n$

8-(iii)

Prove that the only annihilating polynomial of $C$ of degree at most $d-1$ is 0

8-(iv)

Calculate the Jordan normal from form of $C$