Recommended Problems - Rational normal form

Problem 1 (DF, Problem 4, pg. 488)

Prove that two $3\times 3$ matrices over a field $F$ are similar if and only if they have the same characteristic and minimal polynomials. Give an explicit counterexample to this for $4\times 4$ matrices.

Problem 2 (DF, Problem 12,m pg. 489)

Find all similarity classes of $3\times 3$ matrices over $\Z/2\Z$ satisfying $A^6=1$. Do the same for $4\times 4$ matrices satisfying $B^{20}=1$.

Problem 3 (DF, Problem 17, p. 489)

Let $G=\GL_{3}(\mathbf{F}_{2})$. This group has order $(7)(6)(4)=168$. Find representatives for each of its conjugacy classes.

Problem 4 (DF Problems 22-25, pg 490)

These problems show that if $A$ is an $n\times n$ matrix over $F$, then the matrix $xI-A$ is the “relation matrix” for the $F[x]$-module obtained from $F^{n}$ with $x$ acting as $A$. Consequently, to compute the rational normal form for $A$, one must use the “smith normal form” algorithm over $F[x]$ on this matrix to find the invariant factors. (Note: Actually the problem uses $xI-A^{\intercal}$ but this is just a question of conventions regarding rows vs. columns and doesn’t really matter).

Recommended Problems - Jordan normal form

Computations

DF Problems 4-16 on pages 499-501 give a whole bunch of numerical examples; try some.

Problem 5 (DF, Problem 22, page 501)

Prove that an $n\times n$ matrix $A$ over $\mathbf{C}$ satisfying $A^3=A$ is diagonalizable. Is this true for any field $F$?

Problem 6 (DF, Problem 31, pag 502)

A matrix is nilpotent if $A^n=0$ for some $n$. Prove that a nilpotent matrix is similar to a block diagonal matrix where each block has ones on the superdiagonal and zeros elsewhere.

Optional: Problems 40ff in Chapter 12.3 explore applications of the Jordan form to differential equations.

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