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Canonical Forms Problems. Part 1 - nerjik.com

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Canonical Forms Problems. Part 1

Canonical Forms Problems. Part 1

Almaz Khusnutdinov

Almaz Khusnutdinov•Math enthusiast, love coding and mathematics

Published 11/2/2025

Triangular Forms

Problem 1.1

\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]

Problem 1.2

\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]

Problem 1.3

\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]

Problem 1.4

\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]

Problem 1.5

\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]

Problem 1.6

\text{Consider the following upper triangular matrices.}\\[4pt] \text{(a) }A=\begin{bmatrix} 1 & 2 & 0\\ 0 & 3 & 4\\ 0 & 0 & -1 \end{bmatrix}. \ \text{Compute the characteristic polynomial }\Delta_A(t)\text{ and the eigenvalues of }A.\\[4pt] \text{(b) }B=\begin{bmatrix} 2 & 0 & 1\\ 0 & 2 & 3\\ 0 & 0 & 5 \end{bmatrix}. \ \text{Compute the characteristic polynomial }\Delta_B(t)\text{ and the eigenvalues of }B.\\[10pt] \textbf{Detailed solution:}\\ \text{(a) For }A,\ tI-A=\begin{bmatrix} t-1 & -2 & 0\\ 0 & t-3 & -4\\ 0 & 0 & t+1 \end{bmatrix}. \ \text{This matrix is upper triangular, so}\\ \Delta_A(t)=\det(tI-A)=(t-1)(t-3)(t+1).\\ \text{Hence the eigenvalues of }A\text{ are }1,\,3,\,-1.\\[6pt] \text{(b) For }B,\ tI-B=\begin{bmatrix} t-2 & 0 & -1\\ 0 & t-2 & -3\\ 0 & 0 & t-5 \end{bmatrix}, \ \text{which is also upper triangular.}\\ \Delta_B(t)=\det(tI-B)=(t-2)(t-2)(t-5)=(t-2)^2(t-5).\\ \text{Thus the eigenvalues of }B\text{ are }2\text{ (with algebraic multiplicity 2) and }5.\\[6pt] \textbf{Answer: } \Delta_A(t)=(t-1)(t-3)(t+1),\ \lambda_A\in\{1,3,-1\};\quad \Delta_B(t)=(t-2)^2(t-5),\ \lambda_B\in\{2,2,5\}.

Problem 1.7

\text{Let }C=\begin{bmatrix} 4 & -1 & 0\\ 0 & 4 & 2\\ 0 & 0 & 4 \end{bmatrix},\quad D=\begin{bmatrix} 0 & 1 & 0\\ 0 & 5 & 0\\ 0 & 0 & -2 \end{bmatrix}.\\[4pt] \text{(a) Compute the characteristic polynomial and eigenvalues of }C.\\ \text{(b) Compute the characteristic polynomial and eigenvalues of }D.\\[6pt] \textbf{Answer: } \Delta_C(t)=(t-4)^3\ \text{ with eigenvalue }4\text{ of algebraic multiplicity }3;\\ \Delta_D(t)=t(t-5)(t+2)\ \text{ with eigenvalues }0,\,5,\,-2.

Problem 1.8

\text{Consider the triangular matrices}\\[2pt] A=\begin{bmatrix} 1 & 1 & 0\\ 0 & -2 & 3\\ 0 & 0 & 4 \end{bmatrix},\quad B=\begin{bmatrix} -1 & 2\\ 0 & 3 \end{bmatrix}.\\[4pt] \text{(a) For }A,\ \text{find the eigenvalues of }A\text{ and of }A^2\text{ using triangular form.}\\ \text{(b) For }B,\ \text{find the eigenvalues of }B\text{ and of }B^2\text{ using triangular form.}\\[6pt] \textbf{Answer: } \text{For }A:\ \lambda(A)=\{1,-2,4\},\ \lambda(A^2)=\{1,4,16\}.\\ \text{For }B:\ \lambda(B)=\{-1,3\},\ \lambda(B^2)=\{1,9\}.

Problem 1.9

\text{Consider the upper triangular }2\times2\text{ matrices}\\[2pt] \text{(a) }A=\begin{bmatrix} 3 & 1\\ 0 & 3 \end{bmatrix},\quad \text{(b) }B=\begin{bmatrix} a & 2\\ 0 & b \end{bmatrix},\ a,b\in\mathbb{C}.\\[4pt] \text{For each matrix, compute the characteristic polynomial and list its eigenvalues (with multiplicities when appropriate).}\\[6pt] \textbf{Answer: } \Delta_A(t)=(t-3)^2\ \text{ with eigenvalue }3\text{ of algebraic multiplicity }2;\\ \Delta_B(t)=(t-a)(t-b)\ \text{ with eigenvalues }a\text{ and }b.

Problem 1.10

\text{Let }A\text{ be a square matrix over }\mathbb{C}.\\[4pt] \text{(a) Suppose }4\text{ is an eigenvalue of }A^2.\ \text{List all possible eigenvalues }\mu\text{ of }A\text{ such that }\mu^2=4.\\ \text{(b) Suppose }\tfrac{1}{9}\text{ is an eigenvalue of }A^2.\ \text{List all possible eigenvalues }\mu\text{ of }A\text{ such that }\mu^2=\tfrac{1}{9}.\\[6pt] \textbf{Answer: } \text{(a) }\mu=2\ \text{ or }\ \mu=-2;\quad \text{(b) }\mu=\tfrac{1}{3}\ \text{ or }\ \mu=-\tfrac{1}{3}.

Problem 1.11

\text{Consider the following upper triangular matrices.}\\[4pt] \text{(a) }A=\begin{bmatrix} 1 & 2 & 0 & 0\\ 0 & 1 & 3 & 0\\ 0 & 0 & 4 & -1\\ 0 & 0 & 0 & 4 \end{bmatrix}.\ \text{Compute the characteristic polynomial of }A\text{ and list the eigenvalues of }A^2\text{ (with multiplicities).}\\[8pt] \text{(b) }B=\begin{bmatrix} 2 & -1 & 0\\ 0 & 2 & 5\\ 0 & 0 & -3 \end{bmatrix}.\ \text{Compute the characteristic polynomial of }B^2\text{ and list the eigenvalues of }B^2\text{ (with multiplicities).}\\[8pt] \textbf{Answer: } \Delta_A(t)=(t-1)^2(t-4)^2,\ \lambda(A^2)=\{1,1,16,16\};\quad \Delta_{B^2}(t)=(t-4)^2(t-9),\ \lambda(B^2)=\{4,4,9\}.

Problem 1.12

\text{Let }A,B\text{ be }3\times3\text{ complex matrices whose characteristic polynomials factor into linear polynomials.}\\[4pt] \text{(a) Suppose the eigenvalues of }A^2\text{ are }1,9,9\text{ (counted with multiplicity) and }\operatorname{tr}(A)=5.\\ \phantom{\text{(a)}}\text{Find the eigenvalues of }A\text{ (with multiplicities).}\\[4pt] \text{(b) Suppose the eigenvalues of }B^2\text{ are }4,4,25,\ \det(B)=-20,\ \operatorname{tr}(B)=5.\\ \phantom{\text{(b)}}\text{Find the eigenvalues of }B\text{ (with multiplicities).}\\[8pt] \textbf{Answer: } \text{(a) eigenvalues of }A:\ -1,3,3;\quad \text{(b) eigenvalues of }B:\ -2,2,5.

Invariance

Problem 2.1

\text{Let }A,B\text{ be linear operators on }\mathbb{R}^3\text{ with matrices}\\[4pt] A=\begin{bmatrix} 0 & -1 & 0\\ 1 & 0 & 0\\ 0 & 0 & 2 \end{bmatrix}, \qquad B=\begin{bmatrix} 1 & 0 & 0\\ 0 & 2 & 0\\ 0 & 0 & 3 \end{bmatrix}.\\[6pt] \text{(a) Find all subspaces of }\mathbb{R}^3\text{ that are invariant under }A.\\ \text{(b) Find all 1\text{-}dimensional and all 2\text{-}dimensional subspaces of }\mathbb{R}^3\text{ that are invariant under }B.\\[8pt] \textbf{Answer:}\\ \text{(a) Invariant subspaces under }A:\ \{0\},\ \mathbb{R}^3,\ \operatorname{span}\{(0,0,1)\},\ \operatorname{span}\{(1,0,0),(0,1,0)\}.\\[4pt] \text{(b) Invariant subspaces under }B:\\ \quad 1\text{-dimensional: }\operatorname{span}\{(1,0,0)\},\ \operatorname{span}\{(0,1,0)\},\ \operatorname{span}\{(0,0,1)\}.\\ \quad 2\text{-dimensional: }\operatorname{span}\{(1,0,0),(0,1,0)\},\ \operatorname{span}\{(1,0,0),(0,0,1)\},\ \operatorname{span}\{(0,1,0),(0,0,1)\}.

Nilpotent Operators

Problem 3.1

\text{Let }\dim V=7\text{ and }T:V\to V\text{ be nilpotent. Suppose }\\[4pt] \dim(\ker T)=3,\quad \dim(\ker T^2)=5,\quad \dim(\ker T^3)=6,\quad \dim(\ker T^4)=7.\\[10pt] \text{(a) Find the index }k\text{ of }T.\\ \text{(b) Determine the sizes of the Jordan nilpotent blocks of }T\text{ and write a Jordan matrix for }T.\\[14pt] \textbf{Solution: }\\[4pt] \text{(a) Since }\dim(\ker T^4)=7=\dim V,\text{ we have }T^4=0\text{ so }k\le 4.\\ \text{Also }\dim(\ker T^3)=6<7,\text{ so }T^3\ne 0.\text{ Hence }k=4.\\[10pt] \text{(b) For a nilpotent Jordan block }N_s\text{ of size }s,\text{ one has }\\ \dim(\ker N_s^{\,i})=\min(i,s)\quad\Rightarrow\quad \dim(\ker N_s^{\,i})-\dim(\ker N_s^{\,i-1}) =\begin{cases}1,& i\le s,\\ 0,& i>s.\end{cases}\\[10pt] \text{Let }n_i=\dim(\ker T^i)\text{ with }n_0=0.\text{ Define }a_i=n_i-n_{i-1}.\\ \text{Then }a_i\text{ equals the number of Jordan blocks with size }\ge i.\\[10pt] n_0=0,\ n_1=3,\ n_2=5,\ n_3=6,\ n_4=7\ \Rightarrow\\ a_1=3,\ a_2=2,\ a_3=1,\ a_4=1,\ a_5=0.\\[10pt] \text{Now the number of blocks of \emph{exact} size }i\text{ is }a_i-a_{i+1}:\\ \#(4\text{-blocks})=a_4-a_5=1,\quad \#(3\text{-blocks})=a_3-a_4=0,\quad \#(2\text{-blocks})=a_2-a_3=1,\quad \#(1\text{-blocks})=a_1-a_2=1.\\[10pt] \text{So the block sizes are }4,\ 2,\ 1\text{ (they sum to }7\text{), and the total number of blocks is }a_1=3=\dim(\ker T).\\[12pt] \textbf{Answer: }k=4,\ \text{Jordan block sizes }(4,2,1),\\ [T]_J=\operatorname{diag}\!\left( \begin{bmatrix}0&1&0&0\\0&0&1&0\\0&0&0&1\\0&0&0&0\end{bmatrix}, \begin{bmatrix}0&1\\0&0\end{bmatrix}, \begin{bmatrix}0\end{bmatrix} \right).\\ \\[18pt]

Problem 3.2

\text{Let }\dim V=6\text{ and }T:V\to V\text{ be nilpotent with }\\[4pt] \dim(\ker T)=2,\quad \dim(\ker T^2)=4,\quad \dim(\ker T^3)=6.\\[10pt] \text{Find the index }k\text{ and the Jordan nilpotent block sizes of }T.\\[12pt] \textbf{Answer: }k=3,\ \text{block sizes }(3,3),\quad [T]_J=\operatorname{diag}\!\left( \begin{bmatrix}0&1&0\\0&0&1\\0&0&0\end{bmatrix}, \begin{bmatrix}0&1&0\\0&0&1\\0&0&0\end{bmatrix} \right).\\ \\[18pt]

Problem 3.3

\text{Let }\dim V=8\text{ and }T:V\to V\text{ be nilpotent with }\\[4pt] \dim(\ker T)=3\quad\text{and index }k=4.\\[10pt] \text{List all possible multisets of Jordan nilpotent block sizes for }T.\\[12pt] \textbf{Answer: }\{4,3,1\}\ \text{or}\ \{4,2,2\}.\\ \\[18pt]

Problem 3.4

\text{Let }S,T:V\to V\text{ be nilpotent operators on a }7\text{-dimensional space }V\text{ with }\\[4pt] \dim(\ker S)=3,\ \dim(\ker S^2)=5,\ \dim(\ker S^3)=6,\ \dim(\ker S^4)=7,\\ \dim(\ker T)=3,\ \dim(\ker T^2)=5,\ \dim(\ker T^3)=7.\\[10pt] \text{(a) Find the Jordan block sizes (and index) for }S\text{ and for }T.\\ \text{(b) Are }S\text{ and }T\text{ similar?}\\[12pt] \textbf{Answer: }S\text{ has index }4\text{ and block sizes }(4,2,1).\\ T\text{ has index }3\text{ and block sizes }(3,3,1).\\ S\text{ and }T\text{ are not similar.}\\ \\[18pt]

Problem 3.5

\text{Let }\dim V=9\text{ and }T:V\to V\text{ be nilpotent. Suppose}\\[4pt] \dim(\ker T)=4,\quad \dim(\ker T^2)=7,\quad \dim(\ker T^3)=8,\quad \dim(\ker T^4)=9.\\[10pt] \text{(a) Find the index }k\text{ of }T.\\ \text{(b) Determine the Jordan nilpotent block sizes of }T.\\[12pt] \textbf{Answer: }k=4,\ \text{block sizes }(4,2,2,1).\\ \\[18pt]

Problem 3.6

\text{Let }T:\mathbb{R}^6\to\mathbb{R}^6\text{ be defined by the matrix (standard basis)}\\[4pt] A=\begin{bmatrix} 0&1&0&0&0&0\\ 0&0&1&0&0&0\\ 0&0&0&0&0&0\\ 0&0&0&0&1&0\\ 0&0&0&0&0&0\\ 0&0&0&0&0&0 \end{bmatrix}.\\[10pt] \text{(a) Find the nullity }\dim(\ker T)\text{ and the index }k.\\ \text{(b) Give the Jordan nilpotent block sizes of }T.\\[12pt] \textbf{Answer: }\dim(\ker T)=3,\ k=3,\ \text{block sizes }(3,2,1).\\ \\[18pt]

Jordan Canonical Form

Problem 4.1

\text{Let }T:V\to V\text{ be a linear operator with}\\[4pt] \Delta_T(t)=(t-2)^4(t+1)^3,\qquad m_T(t)=(t-2)^3(t+1)^2.\\[10pt] \text{(a) List all possible Jordan block size-multisets for the eigenvalues }2\text{ and }-1.\\ \text{(b) If additionally }\dim\ker(T-2I)=2\text{ and }\dim\ker(T+I)=2,\text{ determine the Jordan form }J\text{ of }T.\\[14pt] \textbf{Solution: }\\[4pt] \text{For an eigenvalue }\lambda,\text{ let }n_\lambda\text{ be the exponent of }(t-\lambda)\text{ in }\Delta_T(t),\\ \text{and let }m_\lambda\text{ be the exponent of }(t-\lambda)\text{ in }m_T(t).\\ \text{Then }n_\lambda\text{ equals the total size of the Jordan blocks for }\lambda,\\ \text{and }m_\lambda\text{ equals the size of the largest Jordan block for }\lambda.\\ \text{Also, the number of Jordan blocks for }\lambda\text{ equals }\dim\ker(T-\lambda I).\\[12pt] \text{Eigenvalue }2:\ n_2=4,\ m_2=3.\\ \text{So the Jordan block sizes for }2\text{ form a partition of }4\text{ whose largest part is }3.\\ \text{The only possibility is }(3,1).\\[10pt] \text{Eigenvalue }-1:\ n_{-1}=3,\ m_{-1}=2.\\ \text{So the Jordan block sizes for }-1\text{ form a partition of }3\text{ whose largest part is }2.\\ \text{The only possibility is }(2,1).\\[12pt] \text{Thus for (a): for }2\text{ we must have }(3,1),\ \text{and for }-1\text{ we must have }(2,1).\\[12pt] \text{For (b): }\dim\ker(T-2I)=2\Rightarrow\text{there are exactly }2\text{ Jordan blocks for }2,\\ \text{which matches the partition }(3,1).\\ \dim\ker(T+I)=2\Rightarrow\text{there are exactly }2\text{ Jordan blocks for }-1,\\ \text{which matches the partition }(2,1).\\[12pt] \text{Therefore }J=\operatorname{diag}\!\big(J_3(2),J_1(2),J_2(-1),J_1(-1)\big).\\[12pt] \textbf{Answer: }\\ \text{(a) For }\lambda=2:\ (3,1)\text{ only;}\quad \text{for }\lambda=-1:\ (2,1)\text{ only.}\\ \text{(b) }J=\operatorname{diag}\!\left( \begin{bmatrix}2&1&0\\0&2&1\\0&0&2\end{bmatrix}, \begin{bmatrix}2\end{bmatrix}, \begin{bmatrix}-1&1\\0&-1\end{bmatrix}, \begin{bmatrix}-1\end{bmatrix} \right).\\ \\[18pt]

Problem 4.2

\text{Let }T:V\to V\text{ have}\\[4pt] \Delta_T(t)=(t-1)^5(t-3)^2,\qquad m_T(t)=(t-1)^2(t-3).\\[10pt] \text{(a) Is }T\text{ diagonalizable?}\\ \text{(b) List all possible Jordan block size-multisets for }T.\\[12pt] \textbf{Answer: }\\ \text{(a) No (since the exponent of }(t-1)\text{ in }m_T\text{ is }2).\\ \text{(b) For }\lambda=1:\ (2,2,1)\text{ or }(2,1,1,1);\quad \text{for }\lambda=3:\ (1,1).\\ \text{So }J=\operatorname{diag}\big(\text{blocks for }1,\ I_2\cdot 3\big)\text{ with the two possibilities above.}\\ \\[18pt]

Problem 4.3

\text{Let }T:V\to V\text{ have}\\[4pt] \Delta_T(t)=t^6(t-4)^3,\qquad m_T(t)=t^3(t-4)^2.\\[10pt] \text{Find all possible values of }\dim\ker T\text{ and the value of }\dim\ker(T-4I).\\[12pt] \textbf{Answer: }\\ \dim\ker(T-4I)=2,\quad \dim\ker T\in\{2,3,4\}.\\ \\[18pt]

Problem 4.4

\text{Let }T:V\to V\text{ have}\\[4pt] \Delta_T(t)=(t-2)^3(t+2)^3,\qquad m_T(t)=(t-2)^2(t+2)^3,\\ \dim\ker(T-2I)=2,\qquad \dim\ker(T+2I)=1.\\[10pt] \text{Determine the Jordan block sizes for each eigenvalue and write a Jordan matrix }J.\\[12pt] \textbf{Answer: }\\ \lambda=2:\ \text{block sizes }(2,1);\quad \lambda=-2:\ \text{block sizes }(3).\\ J=\operatorname{diag}\!\left( \begin{bmatrix}2&1\\0&2\end{bmatrix}, \begin{bmatrix}2\end{bmatrix}, \begin{bmatrix}-2&1&0\\0&-2&1\\0&0&-2\end{bmatrix} \right).\\ \\[18pt]

Problem 4.5

\text{Let }T:V\to V\text{ satisfy}\\[4pt] \Delta_T(t)=(t-1)^6,\qquad m_T(t)=(t-1)^4,\qquad \dim\ker(T-I)=2,\qquad \dim\ker\big((T-I)^2\big)=4.\\[10pt] \text{Determine the Jordan block sizes of }T\text{ and write a Jordan matrix }J.\\[12pt] \textbf{Answer: }\text{block sizes }(4,2),\quad J=\operatorname{diag}\!\left( \begin{bmatrix}1&1&0&0\\0&1&1&0\\0&0&1&1\\0&0&0&1\end{bmatrix}, \begin{bmatrix}1&1\\0&1\end{bmatrix} \right).\\ \\[18pt]

Rational Canonical Form

Problem 5.1

\text{Let }V\text{ be a vector space of dimension }7\text{ over }\mathbb{Q},\text{ and let }T:V\to V\text{ have minimal polynomial}\\[4pt] m_T(t)=(t^3-2)(t-1)^2.\\[10pt] \text{(a) Determine the characteristic polynomial }\Delta_T(t).\\ \text{(b) List all possible rational canonical forms }M\text{ (as block diagonal matrices of companion blocks).}\\[12pt] \textbf{Answer: }\\ \Delta_T(t)=(t^3-2)(t-1)^4.\\[6pt] M=\operatorname{diag}\!\big(C(t^3-2),\,C((t-1)^2),\,C((t-1)^2)\big)\ \text{or}\ M=\operatorname{diag}\!\big(C(t^3-2),\,C((t-1)^2),\,C(t-1),\,C(t-1)\big).\\[10pt] \text{That is, }\\[4pt] C(t^3-2)=\begin{bmatrix}0&0&2\\1&0&0\\0&1&0\end{bmatrix},\quad C((t-1)^2)=\begin{bmatrix}0&-1\\1&2\end{bmatrix},\quad C(t-1)=\begin{bmatrix}1\end{bmatrix}.\\ \\[18pt]

Problem 5.2

\text{Let }V\text{ be a vector space of dimension }8\text{ over }\mathbb{Q},\text{ and let }T:V\to V\text{ have minimal polynomial}\\[4pt] m_T(t)=(t^4+t+1)(t-2)^2.\\[10pt] \text{(a) Determine the characteristic polynomial }\Delta_T(t).\\ \text{(b) List all possible rational canonical forms }M\text{ (as block diagonal matrices of companion blocks).}\\[12pt] \textbf{Answer: }\\ \Delta_T(t)=(t^4+t+1)(t-2)^4.\\[6pt] M=\operatorname{diag}\!\big(C(t^4+t+1),\,C((t-2)^2),\,C((t-2)^2)\big)\ \text{or}\ M=\operatorname{diag}\!\big(C(t^4+t+1),\,C((t-2)^2),\,C(t-2),\,C(t-2)\big).\\[10pt] \text{That is, }\\[4pt] C(t^4+t+1)=\begin{bmatrix} 0&0&0&-1\\ 1&0&0&-1\\ 0&1&0&0\\ 0&0&1&0 \end{bmatrix},\quad C((t-2)^2)=\begin{bmatrix}0&-4\\1&4\end{bmatrix},\quad C(t-2)=\begin{bmatrix}2\end{bmatrix}.\\ \\[18pt]

Problem 5.3

\text{Let }V\text{ be a vector space of dimension }9\text{ over }\mathbb{Q},\text{ and let }T:V\to V\text{ have minimal polynomial}\\[4pt] m_T(t)=(t^2+t+1)^3(t+1).\\[10pt] \text{(a) Determine the characteristic polynomial }\Delta_T(t).\\ \text{(b) Determine the rational canonical form }M\text{ (as a block diagonal matrix of companion blocks).}\\[12pt] \textbf{Answer: }\\ \Delta_T(t)=(t^2+t+1)^4(t+1),\qquad M=\operatorname{diag}\!\big(C((t^2+t+1)^3),\,C(t^2+t+1),\,C(t+1)\big).\\[10pt] \text{That is, }\\[4pt] (t^2+t+1)^3=t^6+3t^5+6t^4+7t^3+6t^2+3t+1,\\ C((t^2+t+1)^3)=\begin{bmatrix} 0&0&0&0&0&-1\\ 1&0&0&0&0&-3\\ 0&1&0&0&0&-6\\ 0&0&1&0&0&-7\\ 0&0&0&1&0&-6\\ 0&0&0&0&1&-3 \end{bmatrix},\quad C(t^2+t+1)=\begin{bmatrix}0&-1\\1&-1\end{bmatrix},\quad C(t+1)=\begin{bmatrix}-1\end{bmatrix}.\\ \\[18pt]

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