3.2 Rational and Jordan canonical form

$T: V \longrightarrow V$ is a fixed linear transformation on a finite-dimensional vector space $V$ over a field $F, A$ is an $n \times n$ matrix over $F, I$ denotes the identity transformation and $I_{n}$ the $n \times n$ identity matrix.

KNOW BASIC DEFINITIONS FOR $T$ (analogously for $A$, interpreted as a linear transformation on $V= F^{n}$ acting by left multiplication): characteristic polynomial $\left(\chi_{T}(x)=\operatorname{det}(x I-T)\right)$, characteristic root (root of the characteristic polynomial, $\lambda$ such that $\operatorname{det}(\lambda I-T)=0$ ), minimum polynomial $\mu_{T}(x)$ (monic polynomial of smallest degree satisfied by $T$ ); eigenvalue (scalar $\lambda \in F$ such that $T(v)=\lambda v$ for some vector $0 \neq v \in V$ ) and eigenvector (vector $0 \neq v \in V$ such that $T(v)=\lambda v$ for some $\lambda \in F) ; \lambda$-eigenspace $V_{\lambda}(=\{v \in V \mid T(v)=\lambda v\}=\{$ eigenvectors of T with eigenvalue $\lambda\} \cup\{0\}) ;$ generalized eigenvector of order $k\left(=\left\{v \in V \mid(T-\lambda I)^{k}(v)=0\right.\right.$ but $\left.(T-\lambda I)^{k-1}(v) \neq 0\right\}$; order 0 means $v=0$, order 1 means $v$ is a true eigenvector); generalized $\lambda$ - eigenspace $V^{\lambda}$ (all generalized eigenvectors); determinant, trace, rank, nullity, Jordan canonical form JCF; $r \times r$ Jordan block $J_{r}(\lambda)$; rational canonical form RCF for a tranformation $T$ or matrix $A$, invariant factors and companion matrices.

Know the following facts:

(1) eigenvalues $=$ roots of the characteristic polynomial
$(\operatorname{det}(\lambda I-T)=0 \Longleftrightarrow \lambda I-T$ singular, kills some vector $v \neq 0, \Longleftrightarrow \lambda v=T(v) \Longleftrightarrow v$ is an eigenvector with eigenvalue $\lambda$ );

(2) minimum polynomial divides all polynomials satisfied by $T$ (it is the generator of the ideal in $F[x]$ of all polynomials $f$ satisfying $f(T)=0$ );

(3) $V_{\lambda}=\operatorname{ker}(T-\lambda I), V^{\lambda}=\bigcup_{k} \operatorname{ker}(T-\lambda I)^{k}$.

(4) $V$ becomes a finitely generated module for the PID $F[x]$ via the action $x . v=T(v)$, so $f(x) . v:= f(T)(v)$; then $V^{\lambda}$ is just the $x-\lambda$-torsion submodule, and a Jordan block $J_{r}(\lambda)$ in JCF corresponds to a cyclic direct summand $F[x] /\left((x-\lambda)^{r}\right)$, in the decomposition of $V$ considered as $F[x]$-module;

(5) every polynomial splits into linear factors over some extension field, for example the algebraic closure of $F$, so we may have to pass to an extension in order to get enough eigenvalues and eigenvectors;

(6) if $d_{1}, \ldots, d_{s}$ are the (monic) invariant factors of $A$ such that $d_{1}\left|d_{2} \ldots\right| d_{s}$, then $\mu_{A}(x)=d_{s}$ and $\chi_{A}(x)=d_{1} \ldots d_{s}$ (implying $\chi_{A}(A)=0$ (Hamilton-Cayley) and also that $\chi_{A}(x)$ divides $\mu_{A}(x)^{n}$ );

(7) the RCF of $A$ is the direct sum (in the sense of matrices) of the $s$ companion matrices of the invariant factors $d_{1}, \ldots, d_{s}$ of $A$.

Know Criterions for Diagonalizability: $T$ is diagonalizable
$\Longleftrightarrow V$ has a basis consisting of eigenvectors of $T$
$\Longleftrightarrow V$ is the (necessarily direct) sum of the eigenspaces $V_{\lambda}$
$\Longleftrightarrow$ for each eigenvalue $\lambda$ of $T, \operatorname{dim}_{F} V_{\lambda}$ equals the multiplicity of $\lambda$ as a root of $\chi_{T}(x)$
$\Longleftrightarrow \chi_{T}(x)$ splits over $F$ and the JCF of $T$ is diagonal
$\Longleftrightarrow \mu_{T}(x)$ is separable and splits over $F$
KNOW how to find $\operatorname{ker}(T)$ (respectively $\operatorname{ker}(T-\lambda I)$ ):
Gaussian reduction on the system $T x=0$, so in particular
$\operatorname{dim} \operatorname{ker}(T)=\#$ free parameters $=\#$ columns $-\#$ leading-one rows $=n-\operatorname{rk}(T)$.

Know How to Count: If a matrix $A$ has characteristic polynomial $\chi(t)=\prod\left(t-\lambda_{i}\right)^{f_{i}}$ and minimum polynomial $\mu(t)=\prod\left(t-\lambda_{i}\right)^{e_{i}}$, then $f_{i} \geq e_{i}$; $e_{i}$ gives the size of the largest Jordan $\lambda_{i^{-}}$ block, $f_{i}$ gives the total sum of the sizes of all Jordan $\lambda_{i}$-blocks. $\operatorname{dim} \operatorname{ker}\left(A-\lambda I_{n}\right)=$ number of Jordan $\lambda$-blocks $=$ number of independent $\lambda$-eigenvectors (exactly one for each Jordan block), $\operatorname{dim} \operatorname{ker}\left(A-\lambda I_{n}\right)^{e}=$ number of independent generalized $\lambda$-eigenvectors of order $\leq e$ (exactly $e$ for each Jordan $\lambda$-block of size $\geq e$, but only $r$ for a Jordan $\lambda$-block of size $r \leq e$ ).

Let $T: V \longrightarrow V$ denote a fixed linear transformation on a finite-dimensional vector space $V$ over a field $F$. We set $n=\operatorname{dim}_{F} V$ and, for any eigenvalue $\lambda$ of $T, N_{\lambda}:=T-\lambda I, k_{e}:=\operatorname{dim}_{F}\left(\operatorname{ker}\left(N_{\lambda}^{e}\right)\right)$ and $r_{e}:=\operatorname{rk}\left(N_{\lambda}^{e}\right)\left(=\operatorname{dim}_{F}\left(\operatorname{im}\left(N_{\lambda}^{e}\right)\right)\right)$. We now count as follows:

For a given eigenvalue $\lambda$ of $T$, the number $l=l(\lambda)$ of Jordan $\lambda$-blocks is
$l=\operatorname{dim} \operatorname{ker}(T-\lambda I)=k_{1}=n-\operatorname{rk}(T-\lambda I)=n-r_{1}$
The number $l_{e}=l_{e}(\lambda)$ of Jordan $\lambda$-blocks of size $e$ is given by the formula
$l_{e}=2 k_{e}-\left(k_{e-1}+k_{e+1}\right)=r_{e-1}-2 r_{e}+r_{e+1}$
Know how to find the JCF and RCF:

(1) Compute the characteristic polynomial,

(2) factor it into irreducibles $x-\lambda_{i}$; FIND JCF FOR EACH EIGENVALUE $\lambda_{i}$ SEPARATELY;

(3) (for roots of multiplicity $m>1$ only) compute $k_{e}=\operatorname{dim}\left(\operatorname{ker}\left(N_{\lambda}^{e}\right)\right)$ for $e=1,2, \ldots m$ (stop when dimension reaches $m ; k_{1}=m$ is the condition that there are enough independent ordinary eigenvectors to diagonalize),

(4) $l_{e}=2 k_{e}-k_{e-1}-k_{e+1}$.

(5) For the RCF, you have to find the invariant factors of $T$. If you already know the JCF, you know the elementary divisors and hence the invariant factors. However, it is sometimes more convenient to derive the invariant factors directly, especially when you know the prime factorization of $\chi_{T}(x)$ in $F[x]$, but not necessarily its roots. For instance, you can then test for which divisors $f$ of $\chi_{T}(x)$ you get $f(T)=0$ in order to determine the minimal polynomial $\mu_{T}(x)$.

Remarks on how to find a Jordan Basis:

(1) This usually involves awkward computations and is therefore not a suitable part of exam problems. It’s different if the given set-up enables you to find a Jordan basis by applying some smart idea instead of boring calculations, e.g. in Problems 17. and 53. below.
Anyhow, the following remarks are sufficient in order to find Jordan bases for "small" matrices $A \in M_{n}(F)$ (or equivalently: $P \in G L_{n}(F)$ such that $P^{-1} A P$ is in JCF ).

(2) If $A$ is diagonalizable, just exhibit a basis consisting of eigenvectors, i.e. solve the homogeneous systems of linear equations yielding $\operatorname{ker}\left(A-\lambda I_{n}\right)$ for each eigenvalue $\lambda$.

(3) If, for a given eigenvalue $\lambda$, there is only one Jordan $\lambda$-block, which must then be of size $m=$ multiplicity of the characteristic root $\lambda$, then choose $v$ in $\operatorname{ker}\left(N_{\lambda}^{m}\right) \backslash \operatorname{ker}\left(N_{\lambda}^{m-1}\right)$, and set $v_{1}=N_{\lambda}^{m-1}(v), v_{2}=N_{\lambda}^{m-2}(v), \ldots, v_{m}=v$, which yields a Jordan basis for this $\lambda$-block.

(4) If there are exactly two Jordan $\lambda$-blocks, and one of them is of size 1 , then first choose $v$ in $\operatorname{ker}\left(N_{\lambda}^{m-1}\right) \backslash \operatorname{ker}\left(N_{\lambda}^{m-2}\right)$, and set $v_{2}=N_{\lambda}^{m-2}(v), v_{3}=N_{\lambda}^{m-3}(v), \ldots, v_{m}=v$. Note that in this case $\operatorname{dim} V_{\lambda}=2$ and $\left\langle v_{2}, \ldots, v_{m}\right\rangle_{F} \cap V_{\lambda}=F v_{2}$. So we may complete the Jordan basis for the $\lambda$-blocks
by choosing $v_{1} \in V_{\lambda} \backslash F v_{2}$.

(5) If there are precisely two Jordan $\lambda$-blocks, and both are of size 2 , then $\operatorname{dim}_{F}\left(V^{\lambda}\right)=4$ and $k_{1}=2=\operatorname{dim}_{F}\left(N_{\lambda}\left(V^{\lambda}\right)\right)$. Now choose a basis $v_{1}, v_{3}$ of $N_{\lambda}\left(V^{\lambda}\right)$ and find $v_{2}, v_{4} \in V^{\lambda}$ such that $v_{1}=N_{\lambda}\left(v_{2}\right)$ and $v_{3}=N_{\lambda}\left(v_{4}\right)$ (the latter amounts to solving inhomogeneous linear equations). It is easily checked that $v_{1}, v_{2}, v_{3}, v_{4}$ is now a Jordan basis for the two $\lambda$-blocks.

Related Problems

In Problems 1. - 15. below, describe (up to similarity, using JCF or RCF) all $n \times n$ matrices $M$ over a specific field $F$ with given characteristic polynomial $\chi(t)$ and/or minimum polynomial $\mu(t)$.

1. (Jan $79 \# 5$ ) $n=n, F=F, \mu(t)=$ product of distinct linear factors (find JCF , find eigenvalues of any polynomial $f(M)$ ).

2. $(\operatorname{Sep} 79 \# 1) n=6, F=\mathbb{C}, \mu(t)=(t+2)^{2}(t-1)(\mathrm{JCF})$.

3. $(\operatorname{Jan} 81 \# 1) n=8, F=\mathbb{C}, \mathbb{R}, \chi(t)=t^{2}\left(t^{4}-1\right)\left(t^{2}-1\right)(\mathrm{RCF}$ over $\mathbb{R}, \mathrm{JCF}$ over $\mathbb{C})$.

4. $(\operatorname{Jan} 82 \# 1 ; \operatorname{Jan} 79 \# 1) n=6, F=\mathbb{C}, \chi(t)=(t+2)^{4}(t-1)^{2}(\mathrm{JCF})$.

5. (Jan 82#3) $n=5, F=\mathbb{C}, \chi(t)=(t-2)^{3}(t+7)^{2}, \mu(t)=(t-2)^{2}(t+7)$ (find trace, determinant, JCF, is it diagonalizable?)

6. $($ Feb $84 \# 4) n=6, F=\mathbb{Q}, \mu(t)=(t-1)^{2}\left(t^{2}+1\right)($ RCF over $\mathbb{Q}, \mathrm{JCF}$ over $\mathbb{C})$.

7. $(\operatorname{Jan} 87 \# 7) n=5, F=\mathbb{C}, \mathbb{R}, \chi(t)=t^{5}-t(\mathrm{RCF}$ over $\mathbb{R}, \mathrm{JCF}$ over $\mathbb{C})$.

8. $(\operatorname{Aug} 88 \# 2) n=6, F=\mathbb{C}, \mathbb{R}, \chi(t)=t^{6}-t^{5}-t^{2}+t(\mathrm{JCF}$ over $\mathbb{C}, \mathrm{RCF}$ over $\mathbb{R})$.

9. $(\operatorname{Jan} 89 \# 5) n=9, F=F, \mu(t)=t^{2}(t-1)^{2}(t+1)^{3}$.

10. (May $91 \# 1 \mathrm{ab}) n=6, F=\mathbb{R}, \mu(t)=t^{4}+t^{2}(\mathrm{RCF}$ over $\mathbb{R}, \mathrm{JCF}$ over $\mathbb{C})$.

11. (May 92 #4) $n=6, F=\mathbb{C}, \mu(t)=(t-1)^{2}(t-2)$ (JCF).

12. (Jan $94 \# 1$ ) $n=5, F=\mathbb{Q}, \mu(t)=(t-2)^{2}(t+3)$ (find all RCFs; find two such matrices having the same characteristic polynomial but which are still not similar).

13. (Jan 95 #4) $n=8, F=\mathbb{C}, \chi(t)=t^{8}-t^{4}, \mu(t)=t^{6}-t^{2}$ (JCF).

14. (Aug 95 Comprehensive #6) $n=10, F=\mathbb{R}, \mu(t)=\left(t^{4}-1\right)^{2}$ (find all possible values of $k$, the maximum number of independent eigenvectors).

15. (Aug 96#7) $n=7, F=\mathbb{C}, \mathbb{R}, \chi(t)=(t-1)^{3}\left(t^{2}+1\right)^{2}, \mu(t)=(t-1)\left(t^{2}+1\right)^{2}$ (RCF over $\mathbb{R}$, $\mathbb{C}, \mathrm{JCF}$ over $\mathbb{C})$.

16. (April $77 \# 1$ ) Use JCF to show $\operatorname{det}\left(e^{M}\right)=e^{\operatorname{trace}(M)}$ for any $n \times n$ complex matrix; find $e^{M}$ for $M=\left(\begin{array}{ll}1 & 1 \\ 0 & 1\end{array}\right)$.

17. (Nov 77 #6) Let $V$ be the real vector space of all polynomials over $\mathbb{R}$ of degree $<n$ and $T=d / d x$ the linear operator on $V$ defined by differentiating. Determine the JCF of $T$ and exhibit a Jordan basis for $T$.

18. (May 78 #7) Find inverse of $\left(\begin{array}{lll}0 & 0 & 1 \\ 0 & 1 & 0 \\ 1 & 0 & 0\end{array}\right)$.

19. (May 78 #8) $F=\mathbb{C}: M=\left(\begin{array}{cccc}0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 1 & 1 & 0 & 0\end{array}\right)$ (find $\chi(t), \mu(t)$, eigenvalues, JCF).

20. (Sep 78 #1) Show that the set of $3 \times 3$ rational matrices commuting with $M=\left(\begin{array}{ccc}0 & 0 & 6 \\ 1 & 0 & 4 \\ 0 & 1 & 2\end{array}\right)$ form a field.

21. (Sep $78 \# 2$ ) $n=12, F=G F(3), \chi(t)=$ product of linear factors; find JCF given $\operatorname{rank}(M) =10, \operatorname{rank}\left(M^{2}\right)=9, \operatorname{rank}\left(M^{3}\right)=9, \operatorname{rank}(M-1)=12, \operatorname{rank}(M-2)=9, \operatorname{rank}\left((M-2)^{2}\right) =7, \operatorname{rank}\left((M-2)^{3}\right)=6$.

22. (May $80 \# 3$ ) $F=\mathbb{C}: M_{1}=\left(\begin{array}{rrrr}1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & -1 & 0 \\ 0 & 0 & 0 & 0\end{array}\right), M_{2}=\left(\begin{array}{rrrr}1 & -1 & 0 & 0 \\ 0 & -1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0\end{array}\right)$ (find JCF and eigenvalues).

23. (Sep $80 \# 3$ ) $n=4, F=\mathbb{C}$, satisfies $f(t)=t^{2}-7 t+10$, trace 11 (find determinant, JCF).

24. (Sep 82 #7) Show two idempotent $n \times n$ matrices are similar iff they are equivalent.

25. (Sep 82 #8) Give two complex $4 \times 4$ matrices that have the same minimum and characteristic polynomials, yet are not similar.

26. (Sep $83 \# 5$ ) If $M$ is any $3 \times 3$ real matrix with characteristic polynomial $x^{3}+a x^{2}+b x+c$, define its adjoint to be $M^{*}:=M^{2}+a M+b I$ and show $M^{*}=\operatorname{det}(M) M^{-1}$ when $M$ is invertible; show $\left(M^{*}\right)^{*}=\operatorname{det}(M) M$ for any $M$.

27. (Sep $83 \# 6$ ) $F=\mathbb{C}: M_{1}=\left(\begin{array}{rrrr}0 & -1 & -1 & -1 \\ 1 & 2 & 1 & 1 \\ 0 & 0 & 0 & -1 \\ 0 & 0 & 1 & 2\end{array}\right), M_{2}=\left(\begin{array}{rrrr}0 & -1 & 0 & -1 \\ 1 & 2 & 1 & 2 \\ 0 & 0 & 0 & -1 \\ 0 & 0 & 1 & 2\end{array}\right)$ (find RCF and JCF).

28. (Sep $84 \# 4$ ) $F=\mathbb{R}, \mathbb{C}: D=d / d x$ on $V=\operatorname{span}\{x \sin (x), x \cos (x), \sin (x), \cos (x)\}$ (find $\chi(t), \mu(t)$, invariant factors, RCF over $\mathbb{R}, \mathrm{JCF}$ over $\mathbb{C})$.

29. (Sep 84 #8) Find the determinant and inverse of the $n \times n$ real matrix

$$\left(\begin{array}{rrrrrr} 1 & 1 & 1 & 1 & \ldots & 1 \\ 1 & 2 & 2 & 2 & \ldots & 2 \\ 1 & 2 & 3 & 3 & \ldots & 3 \\ 1 & 2 & 3 & 4 & \ldots & 4 \\ \ldots & \ldots & \ldots & \ldots & \ldots & \ldots \\ 1 & 2 & 3 & 4 & \ldots & n \end{array}\right)$$

30. $(1985 \# 2) F=\mathbb{R}: M=\left(\begin{array}{rrr}1 & 1 & 1 \\ -1 & -1 & -1 \\ 1 & 1 & 0\end{array}\right)$ (find $\left.\chi(t), \mu(t), \mathrm{JCF}\right)$.

31. (Sep $86 \# 3$ ) Show that linear operators $S, T$ on a finite-dimensional $V$ are similar if (i) $\operatorname{rank}\left(S^{m}\right)=\operatorname{rank}\left(T^{m}\right)$ for all $m=0,1, \ldots$, and (ii) $S^{n}=0$ for some $n$.

32. (Fall $87 \# 6$ ) Find RCF of the $9 \times 9$ matrix over $\mathbb{R}$ whose JCF is $M=\operatorname{Diag}\left\{J_{2}(2), J_{1}(2), J_{2}(2), J_{2}(i), J_{2}(-i)\right\}$.

33. (May $89 \# 1$ ) $F=\mathbb{C}: M=\left(\begin{array}{rrr}0 & 4 & 2 \\ -1 & -4 & -1 \\ 0 & 0 & -2\end{array}\right)$ (find RCF and JCF ).

34. (May 89 #9) If an $n \times n$ matrix $M$ over $F$ has an elementary divisor $\lambda-a(a \in F)$ show there is an invertible $n \times n$ matrix $Q$ over $F$ with $Q^{-1} M Q=\operatorname{RCF}(M)$ AND $\operatorname{det}(Q)=1$.

35. (Aug $89 \# 7$ ) $n=2, F=G F(2)$, ALL POSSIBLE RCFs.

36. (May 1990 #7) Explain which of these $8 \times 8$ matrices are similar.

$$\begin{gathered} M_{1}=\left(\begin{array}{rrrrrrrr} 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 \\ -1 & 2 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 \\ 0 & 0 & -1 & 0 & -3 & 0 & -3 & 0 \end{array}\right), M_{2}=\left(\begin{array}{rrrrrrrr} 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 1 & 1 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & i & 1 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & i & 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & i & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & -i & 1 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & -i & 1 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & -i \end{array}\right), \\ M_{3}=\left(\begin{array}{rrrrrrrr} i & 1 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & i & 1 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & i & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & i & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 & -i & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 1 & -i & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & -i & 1 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 \end{array}\right) . \end{gathered}$$

37. (May $91 \# 1 \mathrm{c})$ Suppose $M$ has the property that $\operatorname{trace}\left(M^{k}\right)=0$ for all $k>0$, and that its minimum polynomial divides $t^{4}+t^{2}$. Show that $M$ is nilpotent.
    (Imprecise formulation of the problem; assume $M \in M_{n}(F)$ and $\operatorname{char}(F)=0$ or $\operatorname{char}(F)>$ n.)

38. (Jan $92 \# 5$ ) $n=n, F=F$, satisfies $f(t)=t^{2}-t$ (is idempotent, $M^{2}=M$ ) (show two such are similar iff they have the same rank).

39. (May $92 \# 7$ ) If a real $2 \times 2$ symmetric matrix $M$ has $\lim _{k \rightarrow \infty} \operatorname{trace}\left(M^{k}\right)=0$, show $\lim _{k \rightarrow \infty} M^{k}=$ 0 too. Does this hold for $n \times n$ real symmetric matrices?

40. (Sep $93 \# 1$ ) $n=n, F=\mathbb{R}$, satisfies $f(t)=t^{3}-t$ (is tripotent, $M^{3}=M$ ) (show two such are similar iff they have the same rank and the same trace).

41. (Aug $94 \# 4 \mathrm{~b}) n=n, F=\mathbb{C}$, satisfies $f(t)=t^{d}-1$ (has order $d$, smallest positive exponent such that $\left.M^{d}=I\right)(\mathrm{JCF})$.

42. (Aug $95 \# 4$ ) If $T$ is a diagonalizable linear operator on a finite-dimensional vector space $V$ and $W$ is a $T$-invariant subspace of $V$, prove $W$ has a $T$-invariant complement (a subspace $U$ such that $V=W \oplus U)$.

43. (Aug $95 \# 5$ ) $F=\mathbb{R}$ : $M=\left(a_{i j}\right)$ with $a_{i j}=1$ for $i \leq j$ and $a_{i j}=0$ for $i>j$ (find JCF).

44. (Jan $97 \# 1) F=\mathbb{R}, \mathbb{C}: M=\left(\begin{array}{rrr}-1 & 3 & 0 \\ 0 & 2 & 0 \\ 2 & 1 & -1\end{array}\right)$ (find RCF over $\mathbb{R}, \mathrm{JCF}$ over $\mathbb{C}$ ).

45. (Aug $97 \# 1 \mathrm{~b}$ ) Let $K$ be a field containing $n$ distinct $n$-th roots of unity, $L$ an extension field such that $\operatorname{Gal}(L / K)$ is a cyclic group of order $n$ with generator $\sigma$. Find the Jordan canonical form of $\sigma$.
    Hint: It is proved in field theory (and you may use this here) that $L$ must be of the form $L=K(\sqrt[n]{a})$ for some $a \in K$.
    Additional question: If $T$ denotes multiplication by $\sqrt[n]{a}$, considered as a $K$-linear transformation of $L$, what is the RCF of $T$ (over $K$ ) and what is the JCF of $T$ (over $L$ )?

46. (Jan $98 \# 7$ ) If the $n \times n$ real matrix $P$ is the transition matrix for a regular Markov chain, then its powers converge to a matrix $T=\lim _{k \rightarrow \infty} P^{k}$ of the form $\left(\begin{array}{rrrr}t_{1} & t_{1} & \ldots & t_{1} \\ t_{2} & t_{2} & \ldots & t_{2} \\ \ldots & \ldots & \ldots & \ldots \\ t_{n} & t_{n} & \ldots & t_{n}\end{array}\right)$ all of whose columns are the same probability vector $\vec{t}$ (its entries $t_{i}$ are all nonnegative and sum to 1 ).
    (a) What is the JCF of the limit $T$ ?
    (b) What are the possible complex eigenvalues of the original $P$ ?
    (c) What are the possible JCFs of $P$ ? If you cannot do the general case, do at least the $2 \times 2$ case.

47. (Aug $98 \# 6$ ) Let $V$ be an $n$-dimensional vector space over the complex numbers, and let $T$ be a linear transformation from $V$ to itself whose minimum polynomial $\mu(x)$ has degree 2 .
    (a) Find all possible Jordan Canonical Forms for $T$. (Hint: consider the possible factorizations of $\mu(x)$ over $K$.)
    (b) Show that $V$ is a direct sum of $T$-invariant subspaces, each of which has dimension less than or equal to 2 .
    (c) Show that $T$ has an eigenvalue $\lambda$ such that the $\lambda$-eigenspace (the set of all eigenvectors for the eigenvalue $\lambda$, together with the zero vector) has dimension at least $n / 2$.

48. (Aug 99 #6) Describe (in terms of Jordan form) all $2 \times 2$ complex matrices which are similar to their square. (Hint: There are more matrices, Horatio, than are dreamt of in your philosophy!)

49. (Aug $99 \# 7$ ) (a) Show that the ring of $2 n \times 2 n$ matrices $M_{2 n}(F)$ over a field $F$ for $n \geq 1$ is "algebraically closed" with respect to polynomials of degree 2 , in the sense that every polynomial $p(x)=x^{2}+\alpha x+\beta \in F[x]$ of degree 2 has a "root" $A \in M_{2 n}(F)$ (in the sense that $p(A)$ is the zero matrix). Can you generalize this to polynomials of degree $d$ ?
    (b) How many real $2 \times 2$ matrices $A \in M_{2}(\mathbb{R})$ are the roots of the polynomial $p(x)=x^{2}+1$ ?

50. (Aug $01 \# 5$ ) How many non-similar linear transformations $T$ can there be on an 8 -dimensional real vector space $V$ having the minimum polynomial $\mu_{T}(t)=(t-2)(t-3)(t-6)^{3}$ ? List their Jordan canonical forms.

51. (Aug $01 \# 6$ ) Consider the linear transformation $T$ on $\mathbb{R}^{3}$ which rotates points around the $z$-axis through a fixed angle $\theta$.
    (a) Find the matrix of $T$ with respect to the canonical basis for $\mathbb{R}^{3}$.
    (b) Find the characteristic polynomial of $T$, and factor it into irreducibles (ignore the cases when $\theta$ is an integer multiple of $\pi$ ).
    (c) Find the Jordan canonical form for $T$ (you may have to pass to complex matrices).

52. (Aug $02 \# 2$ ) Show that if for a matrix $A \in M_{n}(K)$ (where $K$ is a field) there exists a nonzero matrix $B \in M_{n}(K)$ such that $A B=0$, then there also exists a nonzero matrix $C \in M_{n}(K)$ such that $C A=0$ (in other words, in $M_{n}(K)$ the set of left zero divisors coincides with the set of right zero divisors; note that $A B=0$ does not always imply $B A=0$ - find such examples!)

53. (Aug $03 \# 3$ ) Let $P_{2 n-1}$ be the vector space of polynomials in one variable $x$ with real coefficients of degree $\leq 2 n-1$. Let $T$ denote the linear operator on $P_{2 n-1}$ defined by $p(x) \mapsto p(x)+p^{\prime \prime}(x)$ for every $p(x) \in P_{2 n-1}$, where $p^{\prime \prime}$ denotes the second derivative.
    (a)Determine the matrix $A_{T}$ of the operator $T$ with respect to the basis $\left\{1, x, x^{2}, \ldots, x^{2 n-1}\right\}$.
    (b) Find the Jordan canonical form of $A_{T}$ AND a Jordan basis for $T$.

54. (Jan $04 \# 3$ ) Let $A, B \in M_{n}(\mathbb{C})$ be two commuting matrices. Prove that they have a common eigenvector in $\mathbb{C}^{n}$, i.e. there exists a nonzero $v \in \mathbb{C}^{n}$ such that $A v=\lambda v$ and $B v=\mu v$ for some $\lambda, \mu \in \mathbb{C}$.

55. (Jan 04 #6) Prove:
    (a) If $A \in M_{n}(\mathbb{R})$ is idempotent, then it is diagonalizable.
    (b) Two idempotent matrices $A, B \in M_{n}(\mathbb{R})$ are similar if and only if they have the same rank.

56. (Aug 04 #6) Let $K$ be an arbitrary field ( $\rightarrow$ case distinction!). Classify, up to similarity, all matrices $A \in G L_{4}(K)$ of order 2. (Use an appropriate canonical form.)

57. (Aug 04 #7) Consider the group $G=S L_{2}\left(\mathbb{F}_{4}\right)$.
    (a) ( 4 points) Show without specifying any matrix that $G$ contains an element of order 5 .
    (b) ( 8 points) Exhibit a concrete matrix $A \in S L_{2}\left(\mathbb{F}_{4}\right)$ of order 5 . Use $\mathbb{F}_{4}=\left\{0,1, \alpha, \alpha^{2}\right\}$, where $\alpha$ is a root of $x^{2}+x+1$. Describe in detail how you obtained $A$; you shouldn’t just guess! (Hint: You might first factorize $x^{5}-1$ in $\mathbb{F}_{4}[x]$.)

58. (Jan $05 \# 7$ ) Let $A=\left(a_{i j}\right) \in M_{n}(\mathbb{Q})$ be given by $a_{i i}=0$ for all $i$ and $a_{i j}=2$ for all $i \neq j$. Determine the Jordan canonical form of $A$.

59. (Aug 05 #6) Let $A=J_{n}(\lambda) \in M_{n}(\mathbb{C})$ be a Jordan $\lambda$-block $(\lambda \in \mathbb{C})$.
    (a) Determine $\operatorname{dim}_{\mathbb{C}} \mathbb{C}[A]$.
    (b) For $\lambda \neq 0$ and any natural number $k$, determine the Jordan canonical form of $A^{k}$.

Include arguments for both parts!
60. (Aug 05 #7) Show that there does not exist any matrix $A \in M_{10}(\mathbb{Q})$ satisfying $A^{4}=-I_{10}$.
61. (Jan $06 \# 5$ and #6) $V$ is a finite-dimensional $F$-vector space, $F$ a field and $T: V \rightarrow V$ a linear transformation.

(5) Let $T=\lambda I d+Z$ for $\lambda$ in a field $F$ of characteristic 0 and $Z$ nilpotent. Show that if $T^{k}=I d$ for some $k>0$ then $Z$ must be the zero transformation.

(6) If $T(v) \in F v$ for all $v \in V$, show that $T=\lambda I d$ is a "scalar" for some $\lambda \in F$.
62. (Jan 06 #7) Prove: If $G$ is a finite subgroup of $S L_{2}(\mathbb{C})$ then $A v=v$ for $A \in G, v \in \mathbb{C}^{2}$ implies $A=I_{2}$ or $v=0$.
63. (Aug 06 #6) Over $\mathbb{C}$, find the RCF, JCF, the elementary divisors, the invariant factors, the characteristic and the minimal polynomial of:

$$\left(\begin{array}{rrr} 0 & -1 & 3 \\ 1 & 2 & -3 \\ 1 & 1 & -2 \end{array}\right)$$

64. (Jan $08 \# 3$ ) Let $V$ be an $n$-dimensional vector space $V$ over a finite field $F$ of $q$ elements.
    (a) Show that the number of invertible linear operators on $V$ is $\prod_{i=0}^{n-1}\left(q^{n}-q^{i}\right)$.
    (b) Find the cardinality $|P|$ of any 3-Sylow subgroup $P \leq G=\operatorname{GL}\left(4, \mathbb{F}_{81}\right)$ where $n=4$ and $F=\mathbb{F}_{81}$ is the field of $q=81$ elements.

$$|P|=\square$$

(c) Find the cardinality $\left|P^{\prime}\right|$ of any 3-Sylow subgroup $P^{\prime} \leq G^{\prime}$ of the special linear group $G^{\prime}=\operatorname{SL}\left(4, \mathbb{F}_{81}\right)$ (those invertible $4 \times 4$ matrices of determinant 1 ) over a field $\mathbb{F}_{81}$ elements.

$$\left|P^{\prime}\right|=\square$$

(d) Describe up to similarity (in terms of Jordan canonical forms) all possible 3-torsion elements $T$ of the general linear group $\operatorname{GL}\left(4, \mathbb{F}_{81}\right)$ : all invertible operators $T$ with $T^{3^{e}}=I d$ for some $e \geq 0$. For each $T$ list its minimum polynomial $\mu_{T}(x)$ and its 3 -period (the smallest $e \geq 0$ with $T^{3^{e}}=I d$ ). [Hint: all eigenvalues of $T$ already lie in $\mathbb{F}_{3}$.]
65. (Aug 08 #1) Let A be the 5-by-5 real matrix $\mathrm{A}:=\left[\begin{array}{ccccc}2 & 1 & 4 & 0 & 0 \\ 0 & 2 & -1 & 0 & 0 \\ 0 & 0 & 3 & 0 & 0 \\ 0 & 0 & 0 & 3 & 1 \\ 0 & 0 & 0 & 0 & 3\end{array}\right]$
a. What is the characteristic polynomial $\chi_{A}(x)$ and the minimal polynomial $\mu_{A}(x)$ ?
b. Up to similarity, how many matrices $B \in \mathbb{M}_{5 \times 5}(\mathbb{R})$ have the same characteristic polynomial

$$\chi_{B}(x)=\chi_{A}(x) ?$$

c. What is the Jordan Canonical Form of A?
66. (Jan $09 \# 5$ ) Let $F$ be an algebraically closed field, and let $M_{n}(F)$ be the ring of $n \times n$ matrices over $F$.
(a) Prove that for any $A \in M_{n}(F)$ the centralizer of $A$ has dimension at least $n$. (Hint: look at each Jordan block.)
(b) Describe all matrices $A \in M_{n}(F)$ with the property that every matrix commuting with $A$ is diagonalizable.
67. $(\operatorname{Aug} 09 \# 5)$ (a) Let $A=\left(\begin{array}{lll}2 & 1 & 3 \\ 1 & 2 & 4 \\ 0 & 0 & 1\end{array}\right) \in M_{3}(\mathbb{C})$. Find the minimal polynomial, the characteristic polynomial and the Jordan canonical form of $A$.
(b) Let $J_{n}(0) \in M_{n}(\mathbb{C})$ be the Jordan block of size $n \geq 2$ with 0 ’s on the diagonal. Prove that there exists no matrix $A \in M_{n}(\mathbb{C})$ such that $A^{2}=J_{n}(0)$.
68. (Aug $10 \# 5$ ) Let $F$ be a field, $M_{2}(F)$ the ring of $2 \times 2$ matrices over $F$ and $G L_{2}(F)$ the group of invertible elements of $M_{2}(F)$.
(a) Prove that two matrices in $M_{2}(F)$ are similar if and only if they have the same minimal polynomial.
(b) Assume that $F$ is finite, and let $q=|F|$. Find the number of conjugacy classes in $G L_{2}(F)$.
(c) Again assume that $F$ is finite of order $q$. Find the number of nilpotent matrices in $M_{2}(F)$.

Hint: If $A \in M_{2}(F)$ is nilpotent, what is its Jordan normal form? You may use without proof that $\left|G L_{2}(F)\right|=\left(q^{2}-1\right)\left(q^{2}-q\right)$.
69. (Aug 11 #5) This problem tests some standard linear algebra facts. You can quote standard theorems.

Let $F$ be a field and let $T: F^{6} \rightarrow F^{6}$ be a linear operator with characteristic polynomial $\chi_{T}(t)=\left(t^{2}+t+1\right)\left(t^{2}-1\right) t^{2}$.
(a) If $F=\mathbb{R}$, what are the various possibilities for the minimal polynomial $\mu_{T}(t)$ of $T$ ?
(b) Determine $\operatorname{det}(T)$ and $\operatorname{trace}(T)$. Be careful about signs!
(c) Write down the companion matrix $C$ of the polynomial $\chi_{T}(t)$. Calculate the minimal polynomial of $C$.
(d) When $F=\mathbb{C}$, when is $T$ diagonalizable (i.e., represented by a diagonal matrix w.r.t. some basis)? Some explanation in terms of $\chi_{T}(t)$ or $\mu_{T}(t)$ is required.
(e) When $F=\mathbb{R}$, when (if ever) is $T$ represented by a symmetric matrix? Why?
(f) Bonus: Let $S: \mathbb{C}^{m} \rightarrow \mathbb{C}^{m}$ be a nilpotent operator which is represented by a matrix in Jordan normal form having blocks of sizes $\lambda_{1} \geq \lambda_{2} \geq \cdots \geq \lambda_{r} \geq 1$. Let $\lambda^{\prime}=\left(\lambda_{1}^{\prime}, \ldots, \lambda_{s}^{\prime}\right)$ be the partition of $m$ dual (or transpose) to the partition $\lambda=\left(\lambda_{1}, \ldots, \lambda_{r}\right)$ of $m$. What is the significance (in terms of $S$ ) of the integers $\lambda_{i}^{\prime}, i=1, \ldots, s$ ? (Note: your answer should be precisely one [short] sentence! No further explanation is wanted.)
70. (Jan $12 \# 5$ ) Let $F$ be a field, $n$ a positive integer and $M_{n}(F)$ the set of $n \times n$ matrices over $F$. Let $A \in M_{n}(F)$ be such that $A^{2}=A$. Prove that $A$ is diagonalizable and classify all such $A$ up to similarity. (Recall that $A, B \in M_{n}(F)$ are similar if there exists $C \in G L_{n}(F)$ s.t. $C^{-1} A C=B$.)
71. (Aug $12 \# 6$ ) Let $p$ be an odd prime number and $n$ an integer $\geq 2$.
(a) Show that $G L_{n}(\mathbb{Q})$ has an element of order $p$ if and only if $n \geq p-1$.
(b) Show that there is an $A \in G L_{n}(\mathbb{Q})$ of order $p$ which does NOT have 1 as an eigenvalue if and only if $p-1$ divides $n$.
(c) Let $A \in G L_{4}(\mathbb{Q})$ be an element of order 5 . Prove that the complex JCF of $A$ is independent of such $A$ (up to permutation of blocks) and write it down.
72. (Jan $13 \# 6$ ) Let $F$ be an algebraically closed field and $A \in M a t_{n}(F)$ an $n \times n$ matrix over $F$ for some $n \geq 2$.
(a) Prove that there exist a diagonalizable matrix $D$ and a nilpotent matrix $N$ (that is, $N^{k}=0$ for some $k \in \mathbb{N}$ ) such that $A=D+N$ and $D$ and $N$ commute, that is, $D N=N D$.
(b) Assume that $A$ itself is diagonalizable. Prove that if $D$ and $N$ satisfy the conditions of part (a), then $N=0$ (and hence $D=A$ ). Hint: You may use the following fact without proof: if two diagonalizable matrices $X$ and $Y$ commute, then they are simultaneously diagonalizable, that is, there exists an invertible matrix $Q$ such that $Q^{-1} X Q$ and $Q^{-1} Y Q$ are both diagonal.
73. (Aug 13 #4) Let $K$ be a field, and let $M_{n}(K)$ be the ring of $n \times n$ matrices with entries in $K$. For this problem, let $D \in M_{n}(K)$ be diagonalizable (over $K$ ) and, for each eigenvalue $\lambda$ of $D$, let

$$E_{\lambda}:=\left\{v \in K^{n} \mid D v=\lambda v\right\}$$

be the corresponding eigenspace.
(a) For any $A \in M_{n}(K)$, show that $A D=D A$ if and only if $A\left(E_{\lambda}\right) \subseteq E_{\lambda}$ for all eigenvalues $\lambda$ of $D$.
(Hint: For the "if" part, you may use that $A D=D A$ if $A D v=D A v$ for all $v \in K^{n}$.)
(b) If $A$ is also diagonalizable and $A D=D A$, show that $A$ and $D$ are simultaneously diagonalizable (that is, there is a matrix $P$ such that both $P A P^{-1}$ and $P D P^{-1}$ are diagonal). Provide a counter-example showing that this need not be the case if the matrices do not commute.
(c) If $D$ is invertible, show that the centralizer of $D$ in $G L_{n}(K)$ is isomorphic to a direct product $G L_{n_{1}}(K) \times \ldots \times G L_{n_{r}}(K)$, where $n_{1}+\ldots+n_{r}=n$. Also show that each of these products can be realized as the centralizer of some (appropriately chosen) $D$, provided that $K$ has at least $n+1$ elements.
74. (Jan $14 \# 1$ ) Let $K$ be an algebraically closed field of characteristic 0 . Show that any element of finite order in $G L_{n}(K)$ is diagonalizable. (Hint: Jordan Form!).
75. (Aug $14 \# 4$ ) Denote by $J$ the $n \times n$ Jordan block with eigenvalue 0 . For a positive integer $k$, determine the Jordan canonical form of $J^{k}$. (Hint: you can start by playing with some small values of $k$ ).
76. (Aug 15 #4) Find the characteristic polynomial, the minimal polynomial, and the Jordan canonical form of the matrix (over the complex numbers)

$$A=\left(\begin{array}{cccc} 1 & 2 & 0 & 1 \\ 1 & 0 & 0 & -1 \\ 0 & 0 & 2 & 0 \\ -1 & 0 & 0 & 1 \end{array}\right)$$

77. (Aug $15 \# 6$ ) Let $F$ be a field and $A$ an $n$ by $n$ matrix with coefficients in $F$. Assume that $A$ has only one invariant factor. Prove that for every $n$ by $n$ matrix $B$ with coefficients in $F$ such that $A B=B A$ there is a polynomial $p(t) \in F[t]$ such that $p(A)=B$. (Hint: consider the structure of $V=F^{n}$ as an $F[A]$-module. Use that an endomorphism is determined by its action on a basis.)

78. (Jan $16 \# 5$ ) Let $T$ be a linear operator on a finite dimensional vector space $V$ over a field $F$. Prove that