Exercise 6.4.1

Let $V$ be a finite-dimensional vector space. What is the minimal polynomial for the identity operator on $V$? What is the minimal polynomial for the zero operator?

Solution: The minimal polynomial for the identity operator is $x-1$. It annihilates the identity operator and the monic zero degree polynomial $p(x)=1$ does not, so it must be the minimal polynomial. The minimal polynomial for the zero operator is $x$. It is a monic polynomial that annihilates the zero operator and again the monic zero degree polynomial $p(x)=1$ does not, so it must be the minimal polynomial.

Exercise 6.4.2

Let $a,b$ and $c$ be tlements of a field $F$, and let $A$ be the following $3\times3$ matrix over $F$:

$$A=\left[\begin{array}{ccc}0 & 0 & c\\1 & 0 & b\\ 0 & 1 & a\end{array}\right].$$Prove that the characteristic polynomial for $A$ is $x^x-ax^2-bx-c$ and that this is also the minimal polynomial for $A$.

Solution: The characteristic polynomial is

$$\left|\begin{array}{ccc} x & 0 & -c\\-1 & x & -b\\0 & -1 & x-a\end{array}\right|=\left|\begin{array}{ccc} x & 0 & -c\\-1 & 0 & x^2-ax-b\\0 & -1 & x-a\end{array}\right|$$$$=1\cdot\left|\begin{array}{cc}x & -c\\ -1 & x^2-ax-b\end{array}\right|=x^3-ax^2-bx-c.$$Now for any $r,s\in F$

$$A^2+rA+s=\left[\begin{array}{ccc}0 & c & ac\\0 & b & c+ba\\1 & a & b+a^2\end{array}\right] + \left[\begin{array}{ccc}0 & 0 & rc\\r& 0 &rb\\0 & r& ra\end{array}\right] + \left[\begin{array}{ccc}s & 0& 0\\0 & s & 0\\0 & 0 & s\end{array}\right]$$$$=\left[\begin{array}{ccc}s & c & ac+rc\\r & b +s& c+ba+br\\1 & a+r & b+a^2+ra+s\end{array}\right]\not=0.$$Thus $f(A)\not=0$ for all $f\in F[x]$ such that $\deg(F)=2$. Thus the minimum polynomial cannot have degree two, it must therefore have degree three. Since it divides $x^3-ax^2-bx-c$ it must equal $x^3-ax^2-bx-c$.

Exercise 6.4.3

Let $A$ be the $4\times4$ real matrix

$$\left[\begin{array}{cccc}1 & 1 & 0 & 0\\

-1 & -1 & 0 & 0\\

-2 & -2 & 2 & 1\\

1 & 1 & -1 & 0\end{array}\right].$$Show that the characteristic polynomial for $A$ is $x^2(x-1)^2$ and that it is also the minimal polynomial.

Solution: The characteristic polynomial equals

$$\left|\begin{array}{cccc}x-1 & -1 & 0 & 0\\

1 &x+ 1 & 0 & 0\\

2 & 2 & x-2 & -1\\

-1 & -1 & 1 & x\end{array}\right|=\left|\begin{array}{cc}x-1&-1\\1&x+1\end{array}\right|\cdot\left|\begin{array}{cc}x-2&-1\\1&x\end{array}\right|,$$$$=x^2(x^2-2x+1)=x^2(x-1)^2.$$Here we used (5-20) page 158. The minimum polynomial is clearly not linear, thus the minimal polynomial is one of $x^2(x-1)^2$, $x^2(x-1)$, $x(x-1)^2$ or $x(x-1)$. We will plug $A$ in to the first three and show it is not zero. It will follow that the minimum polynomial must be $x^2(x-1)^2$.

$$A^2=

\left[\begin{array}{cccc}

0 & 0 & 0 & 0\\

0 & 0 & 0 & 0\\

-3 & -3 & 3 & 2\\

2 & 2 & -2 & -1

\end{array}\right]$$$$A-I=

\left[\begin{array}{cccc}

0 & 1 & 0 & 0\\

-1 & -2 & 0 & 0\\

-2 & -2 & 1 & 1\\

1 & 1 & -1 & -1

\end{array}\right]$$and

$$(A-I)^2=

\left[\begin{array}{cccc}

-1 & -2 & 0 & 0\\

2 & 3 & 0 & 0\\

1 & 1 & 0 & 0\\

0 & 0 & 0 & 0

\end{array}\right]$$Thus

$$A^2(A-I)=

\left[\begin{array}{cccc}

0 & 0 & 0 & 0\\

0 & 0 & 0 & 0\\

-1 & -1 & 1 & 1\\

1 & 1 & -1 & -1

\end{array}\right]\not=0$$$$A(A-I)^2=

\left[\begin{array}{cccc}

1 & 1 & 0 & 0\\

-1 & -1& 0 & 0\\

0 & 0 & 0 & 0\\

0& 0& 0 & 0

\end{array}\right]\not=0$$and

$$A(A-I)=

\left[\begin{array}{cccc}

-1 & -1 & 0 & 0\\

1 & 1& 0 & 0\\

-1 & -1 & 1 & 1\\

1& 1& -2 & -2

\end{array}\right]\not=0.$$Thus the minimal polynomial must be $x^2(x-1)^2$.

Exercise 6.4.4

Is the matrix $A$ of Exercise 3 similar over the field of complex numbers to a diagonal matrix?

Solution: Not diagonalizable, because for characteristic value $c=0$ the matrix $A-cI=A$ and $A$ is row equivalent to

$$\left[\begin{array}{cccc}1&1&0&0\\0&0&1&0\\0&0&0&1\\0&0&0&0\end{array}\right]$$which has rank three. So the null space has dimension one. So if $W$ is the null space for $A-cI$ then $W$ has dimension one, which is less than the power of $x$ in the characteristic polynomial. So by Theorem 2, page 187, $A$ is not diagonalizable.

Exercise 6.4.5

Let $V$ be an $n$-dimensional vector space and let $T$ be a linear operator on $V$. Suppose that there exists some positive integer $k$ so that $T^k=0$. Prove tht $T^n=0$.

Solution: $T^k=0$ $\Rightarrow$ the only characteristic value is zero. We know the minimal polynomial divides this so the minimal polynomial is of the form $t^r$ for some $1\leq r\leq n$. Thus by Theorem 3, page 193, the characteristic polynomial’s only root is zero, and the characteristic polynomial has degree $n$. So the characteristic polynomial equals $t^n$. By Theorem 4 (Caley-Hamilton) $T^n=0$.

Exercise 6.4.6

Find a $3\times3$ matrix for which the minimal polynomial is $x^2$.

Solution: If $A^2=0$ and $A\not=0$ then the minimal polynomial is $x$ or $x^2$. So any $A\not=0$ such that $A^2=0$ has minimal polynomial $x^2$. E.g.

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

Exercise 6.4.7

Let $n$ be a positive integer, and let $V$ be the space of polynomials over $\Bbb R$ which have degree at most $n$ (throw in the $0$-polynomial). Let $D$ be the differentiation operator on $V$. What is the minimal polynomial for $D$?

Solution: $1,x,x^2,\dots,x^n$ is a basis.

$$1\mapsto0$$$$x\mapsto1$$$$x^2\mapsto2x$$$$\vdots$$$$\rule{4mm}{0mm}x^n\mapsto nx^{n-1}$$The matrix for $D$ is therefore

$$\left[\begin{array}{cccccc}

0 & 1 & 0 & 0 & \cdots & 0\\

0 & 0 & 2 & 0 & \cdots & 0\\

0 & 0 & 0 & 3 & \cdots & 0\\

\vdots & \vdots & \vdots & \vdots & \ddots & \vdots\\

0 & 0 & 0 & 0 & \cdots &n\end{array}\right]$$Suppose $A$ is a matrix such that $a_{ij}=0$ except when $j=i+1$. Then $A^2$ has $a_{ij}=0$ except when $j=i+2$. $A^3$ has $a_{ij}=0$ except when $j=i+3$. Etc., where finally $A^n=0$. Thus if $a_{ij}\not=0$ $\forall$ $j=i+1$ then $A^k\not=0$ for $k<n$ and $A^n=0$. Thus the minimum polynomial divides $x^n$ and cannot be $x^k$ for $k<n$. Thus the minimum polynomial is $x^n$.

Exercise 6.4.8

Let $P$ be the operator on $\mathbb R^2$ which projects each vector onto the $x$-axis, parallel to the $y$-axis: $P(x,y)=(x,0)$. Show that $P$ is linear. What is the minimal polynomial for $P$?

Solution: $P$ can be given in the standard basis by left multiplication by $A=\left[\begin{array}{cc}1&0\\0&0\end{array}\right]$. Since $P$ is given by left multiplication by a matrix, $P$ is clearly linear. Since $A$ is diagonal, the characteristic values are the diagonal values. Thus the characteristic values of $A$ are $0$ and $1$. The characteristic polynomial is a degree two monic polynomial for which both $0$ and $1$ are roots. Therefore the characteristic polynomial is $x(x-1)$. If the characteristic polynomial is a product of distinct linear terms then it must equal the minimal polynomial. Thus the minimal polynomial is also $x(x-1)$.

Exercise 6.4.9

Let $A$ be an $n\times n$ matrix with characteristic polynomial

$$f=(x-c_1)^{d_1}\cdots(x-c_k)^{d_k}.$$Show that

$$c_1d_1+\cdots+c_kd_k=\text{trace}(A).$$Solution: Suppose $A$ is $n\times n$. Claim: $|xI-A|=x^n+\text{trace}(A)x^{n-1}+\cdots$. Proof by induction: case $n=2$. $A=\left[\begin{array}{cc}a&b\\c&d\end{array}\right]$. $|xI-A|=x^2-(a+d)x+(ad-bc)$. The trace of $A$ is $a+d$ so we have established the claim for the case $n=2$. Suppose true for up to $n-1$. Let $r=a_{22}+a_{33}+\cdots+a_{nn}$. Then

$$\left|\begin{array}{cccc}

x-a_{11} & a_{12} & \cdots & a_{1n}\\

a_{21} & x-a_{22} & \cdots & a_{2n}\\

\ddots & \ddots & \cdots & \ddots\\

a_{n1} & a_{n2} & \cdots & x-a_{nn}

\end{array}\right|$$Now expanding by minors using the first column, and using induction, we get that this equals

$$(x-a_{11})(x^{n-1}-rx^{n-2}+\cdots)$$$$-a_{21}(\text{polynomial of degree $n-2$})$$$$+a_{31}(\text{polynomial of degree $n-2$})$$$$+\cdots$$$$=x^n+(r+a_{11})x^{n-1}+\text{polynomial of degree at most $n-2$}$$$$=x^n-\text{tr$(A)$}x^{n-1} +\cdots$$Now if $f(x)=(x-c_1)^{d_1}\cdots(x-c_k)^{d_k}$ then the coefficient of $x^{n-1}$ is $c_1d_1+\cdots c_kd_k$ so it must be that $c_1d_1+\cdots c_kd_k=\text{tr}(A)$.

Exercise 6.4.10

Let $V$ be the vector space of $n\times n$ matrices over the field $F$. Let $A$ be a fixed $n\times n$ matrix. Let $T$ be the linear operator on $V$ defined by

$$T(B)=AB.$$Show that the minimal polynomial for $T$ is the minimal polynnomial for $A$.

Solution: If we represent a $n\times n$ matrix as a column vector by stacking the columns of the matrix on top of each other, with the first column on the top, then the transformation $T$ is represented in the standard basis by the matrix

$$M=\left[\begin{array}{cccc}A & & & \\& A & \Large 0 & \\ & \Large 0 & \ddots & \\ & & & A\end{array}\right].$$And since

$$f(M)=\left[\begin{array}{cccc}f(A) & & & \\& f(A) & \Large 0 & \\ & \Large 0 & \ddots & \\ & & & f(A)\end{array}\right]$$it is evident that $f(M)=0$ $\Leftrightarrow$ $f(A)=0$.

Exercise 6.4.11

Let $A$ and $B$ be $n\times n$ matrices over the field $F$. According to Exercise 9 of Section 6.2, the matrices $AB$ and $BA$ have the same characteristic values. Do they have the same characteristic polynomial? Do they have the same minimal polynomial?

Solution: In Exercise 9 Section 6.2 we showed $|xI=AB|=0$ $\Leftrightarrow$ $|xI-BA|=0$. Thus we have two monic polynomials of degree $n$ with exactly the same roots. Thuse they are equal. So the characteristic polynomials are equal. But the minimum polynomials need not be equal. To see this let $A=\left[\begin{array}{cc}0&0\\1&0\end{array}\right]$ and $B=\left[\begin{array}{cc}1&0\\0&0\end{array}\right]$. Then $AB=\left[\begin{array}{cc}0&0\\1&0\end{array}\right]$ and $BA=\left[\begin{array}{cc}0&0\\0&0\end{array}\right]$ so the minimal polynomial of $BA$ is $x$ and the minimal polynomial of $AB$ is clearly not $x$ (it is in fact $x^2$).