Canonical Forms Problems. Part 1

Published 11/2/2025

\text{Let }A=\begin{bmatrix}2&1&0\\[2pt]0&3&4\\[2pt]0&0&5\end{bmatrix}. \quad\text{(a) Compute }\Delta(t)=\det(tI-A).\quad \text{(b) Find the eigenvalues and verify that }\operatorname{tr}(A),\det(A)\text{ equal the sum/product of eigenvalues.}\\[8pt] \textbf{Detailed solution: }tI-A=\begin{bmatrix}t-2&-1&0\\[2pt]0&t-3&-4\\[2pt]0&0&t-5\end{bmatrix}\ \text{is upper triangular, so}\\[4pt] \Delta(t)=\det(tI-A)=(t-2)(t-3)(t-5).\\[4pt] \text{Eigenvalues: }2,3,5.\quad \operatorname{tr}(A)=2+3+5=10,\ \det(A)=2\cdot3\cdot5=30.\\[4pt] \text{Sum/product of eigenvalues: }2+3+5=10,\ \ 2\cdot3\cdot5=30.\ \checkmark\\[8pt] \textbf{Answer: }\Delta(t)=(t-2)(t-3)(t-5),\ \ \text{eigenvalues }2,3,5,\ \ \operatorname{tr}(A)=10,\ \det(A)=30.\\[12pt]

\text{Prove (outline): If }A\text{ is triangular, then }\Delta(t)=\det(tI-A)=\prod_{i=1}^n (t-a_{ii}).\\[6pt] \textbf{Answer (outline): } \text{(1) }tI-A\ \text{is triangular with diagonal entries }t-a_{11},\dots,t-a_{nn}.\\ \text{(2) The determinant of a triangular matrix is the product of its diagonal entries.}\\ \text{(3) Hence }\det(tI-A)=(t-a_{11})\cdots(t-a_{nn}).\\[10pt]

\text{Prove (outline): If the characteristic polynomial of }T:V\to V\ \text{splits over }\mathbb{F},\ \text{then there exists a basis in which }[T]\ \text{is upper triangular.}\\[6pt] \textbf{Answer (outline): } \text{(1) Splitting }\Rightarrow\ \text{there exists an eigenvalue }\lambda\ \text{and eigenvector }v_1.\\ \text{(2) Extend }v_1\ \text{to a basis }B=(v_1,\dots,v_n).\ \text{In this basis, }[T]_B=\begin{bmatrix}\lambda&*\\0&*\end{bmatrix}\ \text{(block upper triangular).}\\ \text{(3) The span of }v_1^\perp\ \text{(the remaining }n-1\text{ vectors) is }T\text{-invariant; restrict }T\ \text{to it.}\\ \text{(4) Apply induction on dimension to triangularize the restriction; combine blocks to get an upper triangular }[T].\\[10pt]

\text{Let }A=\begin{bmatrix}1&1&0\\[2pt]0&1&0\\[2pt]0&0&2\end{bmatrix}. \ \text{Find the minimal polynomial }m_A(t).\\[6pt] \textbf{Answer: }m_A(t)=(t-1)^2(t-2).\\[10pt]

\text{Let }A=\begin{bmatrix}a&*&*\\[2pt]0&b&*\\[2pt]0&0&c\end{bmatrix}\ \text{be upper triangular (stars arbitrary), and let }m\in\mathbb{N}.\\ \text{Compute }\det(A^m)\ \text{and }\operatorname{tr}(A^m).\\[6pt] \textbf{Answer: }\det(A^m)=(abc)^m,\qquad \operatorname{tr}(A^m)=a^m+b^m+c^m.\\[10pt]