-
Consider the matrix
$$
A=\pmatrix{
0 & 2 & -1 \\
-2 & 3 & -2 \\
-3 & 2 & -2
}$$
- Find the characteristic polynomial $\chi_A(x)$ and show it is of the form $-\left(x-λ_1\right)\left(x-λ_2\right)^2$ for some $λ_1 \neq λ_2$.
- Find basis vectors $u$ of $\ker\left(A-λ_1 I\right), v_1$ of $\ker\left(A-λ_2 I\right)$ and $v_1, v_2$ of $\ker\left(A-λ_2 I\right)^2$
- Explain why $\left(A-λ_2 I\right)v_2$ is a scalar multiple of $v_1$.
- Find the matrix of the linear transformation $A$ with respect to the new basis $v_1, v_2, u$.
- $\chi_A(x)=\det(xI-A)=-(x+1)(x-1)^2⇒λ_1=-1,λ_2=1$
- $A-λ_1I=A+I=\pmatrix{
1 & 2 & -1 \\
-2 & 4 & -2 \\
-3 & 2 & -1
}⇒\ker\left(A-λ_1 I\right)=⟨u⟩,u=\pmatrix{0\\1\\2}$
$A-λ_2I=A-I=\pmatrix{ -1 & 2 & -1 \\ -2 & 2 & -2 \\ -3 & 2 & -3}⇒\ker\left(A-λ_2 I\right)=⟨v_1⟩,v_1=\pmatrix{1\\0\\-1}$
$(A-λ_2I)^2=\pmatrix{ 0 & 0 & 0 \\ 4 & -4 & 4 \\ 8 & -8 & 8}⇒\ker\left(A-λ_2 I\right)^2=⟨v_1,v_2⟩,v_2=\pmatrix{1\\1\\0}$ - $v_2∈\ker\left(A-λ_2 I\right)^2⇒\left(A-λ_2 I\right)\big(\left(A-λ_2 I\right)v_2\big)=0⇒\left(A-λ_2 I\right)v_2∈\ker\left(A-λ_2 I\right)⇒\left(A-λ_2 I\right)v_2$ is a scalar multiple of $v_1$.
- $Av_1=v_1,Av_2=v_1+v_2,Au=-u$, so the matrix of $A$ with respect to the basis $v_1,v_2,u$ is $\pmatrix{1&1&0\\0&1&0\\0&0&-1}$
Calculate inverse matrix: $P=(v_1,v_2,u)=\pmatrix{1&1&0\\0&1&1\\-1&0&2},P^{-1}AP=\pmatrix{1&1&0\\0&1&0\\0&0&-1}$
- Write down all possible Jordan normal forms for matrices with characteristic polynomial $(x-λ)^5$. In each case, calculate the minimal polynomial and the geometric multiplicity of the eigenvalue $λ$. Verify that this information determines the Jordan normal form for this choice of characteristic polynomial.
Solution.
There are 7 partitions of 5 into positive integers, up to permutation.
No two rows have identical entries in both the second and the third column, so this information determines the Jordan normal form for this choice of characteristic polynomial.size of Jordan blocks minimal polynomial geometric multiplicity of λ 5 $(x-λ)^5$ 1 4+1 $(x-λ)^4$ 2 3+2+1 $(x-λ)^3$ 3 3+1+1+1 $(x-λ)^3$ 4 2+2+1 $(x-λ)^2$ 3 2+1+1+1 $(x-λ)^2$ 4 1+1+1+1+1 $x-λ$ 5 -
Prove that every square matrix over the complex numbers is conjugate to its transpose, i.e. prove that given any $(n × n)$-matrix $A$ there exists an $(n × n)$-matrix $P$ such that $P^{-1} A P=A^{\sf T}$ where $A^{\sf T}$ is the transpose of $A$.
Proof.
Let the Jordan form of $A$ be $J$, then $A$ is conjugate to $J$. Let the Jordan blocks of $J$ be $J_1,…,J_ℓ$. For each $i=1,…,l$, let $$B_i = \begin{bmatrix}&&&1 \\&&1\\&⋰\\1\end{bmatrix} \qquad \text{and} \qquad J_i = \begin{bmatrix}λ&1\\&⋱&⋱\\&&λ&1\\&&&λ\end{bmatrix}$$ Conjugation by $B_i$ is flipping the matrix vertically and horizontally, which is equivalent to rotating the matrix 180°. Rotate $J_i^{\sf T}$ we get $J_i$, so $B_i^{-1}J_i^{\sf T}B_i=J_i$. Let $B=\operatorname{diag}(B_1,B_2,…,B_ℓ)$. Then $B^{-1}J^{\sf T}B=J$. Therefore, $J$ is conjugate to $J^{\sf T}$. Also $J^{\sf T}$ is conjugate to $A^{\sf T}$, by transitivity $A$ is conjugate to $A^{\sf T}$. - Let $\left\{e_1, e_2, e_3\right\}$ be the usual basis $\left\{(1,0,0)^{\sf T},(0,1,0)^{\sf T},(0,0,1)^{\sf T}\right\}$ of $\mathbb{R}^3$. Express the dual basis to $\left\{(1,0,0)^{\sf T},(1,-1,1)^{\sf T},(2,-4,7)^{\sf T}\right\}$ in terms of $e_1', e_2', e_3'$.
Proof.
Let $\{v_1,v_2,v_3\}$ be a basis. To find the dual basis $\{v_1',v_2',v_3'\}$, we need $v_i'(v_j)=δ_{ij}⇔\pmatrix{v_1'\\v_2'\\v_3'}\pmatrix{v_1&v_2&v_3}=I$. So we invert the matrix $\pmatrix{v_1&v_2&v_3}$ and read by row. $$\pmatrix{1 & 1 & 2 \\0 & -1 & -4 \\0 & 1 & 7}^{-1}=\pmatrix{ 1 & \frac{5}{3} & \frac{2}{3} \\ 0 & -\frac{7}{3} & -\frac{4}{3} \\ 0 & \frac{1}{3} & \frac{1}{3}}$$So the dual basis to $\left\{(1,0,0)^{\sf T},(1,-1,1)^{\sf T},(2,-4,7)^{\sf T}\right\}$ is $\left\{e_1'+\frac53e_2'+\frac23e_3',-\frac73e_2'-\frac43e_3',\frac13e_2'+\frac13e_3'\right\}$. -
Let $T:V→W$ be a linear map between finite-dimensional vector spaces. Prove that
$$
\operatorname{Im}\left(T'\right)=(\ker T)^0.
$$
Proof.
$∀f∈W',\ v ∈\ker T:Tv=0⇒T'(f)(v)=f(Tv)=f(0)=0 ⇒T'(f)∈(\ker T)^0$. Therefore $\operatorname{Im}\left(T'\right)⊂(\ker T)^0$.
$∀f∈(\ker T)^0$, the quotient map $\bar f∈(V/\ker T)'$ is given by $\bar f(v+\ker T)=f(v)$. By first isomorphism theorem, we have an isomorphism $g:\operatorname{Im}T→V/\ker T$.
Then $\bar f(g(T(v)))=f(v)⇒f=T'(\bar f∘g)∈\operatorname{Im}\left(T'\right)$. Therefore $\operatorname{Im}\left(T'\right)⊃(\ker T)^0$. - Let $U$ be a subspace of $V$. Show that restriction $f↦\left.f\right|_U$ defines a linear map $V' → U'$. Deduce that there is a natural injection $V' / U^0 → U'$ which is an isomorphism when $V$ is finite-dimensional.
Proof.
$∀f,g ∈ V',\ u∈U,\ \left.(f+λg)\right|_U(u)=f(u)+λg(u)=\left(\left.f\right|_U+λ\left.g\right|_U\right)(u)⇒\left.(f+λg)\right|_U=\left.f\right|_U+λ\left.g\right|_U$, so $f↦\left.f\right|_U$ is a linear map. By definition the kernel is $U^0$. By the first isomorphism theorem, the image is isomorphic to $V'/U^0$, thus giving us an injection $V'/U^0\to U'$.
Let $V$ be finite dimensional. For any $ψ∈U'$, define $f∈V'$ as $f(v)=ψ(\text{proj}_Uv)$, then $\left.f\right|_U=ψ$. We conclude that the restriction is surjective. -
(i) Let $V$ be a finite dimensional vector space over $𝔽$. For a linear transformation $T: V → V$ define the trace $\operatorname{tr}(T)$ to be the trace of the matrix representing $T$ with respect to some basis of $V$. Show that $\operatorname{tr}(T)$ is well-defined, i.e. show that it is independent of choice of basis.
(ii) As usual let $\operatorname{Hom}(V, V)$ denote the space of linear maps from $V$ to itself. For $S∈\operatorname{Hom}(V, V)$ define $f_S:\operatorname{Hom}(V, V)→𝔽$ by $$ f_S: T↦\operatorname{tr}(S∘T) $$ Show that $S↦f_S$ defines a linear isomorphism between $\operatorname{Hom}(V, V)$ and its dual that does not depend on a choice of basis. Solution.
(i) Let $M$ be the matrix of $T$ with respect to some basis of $V$. Then the matrix of $T$ with respect to any basis of $V$ is of the form $P^{-1}MP$. By the cyclic property of trace, $\operatorname{tr}(P^{-1}MP)=\operatorname{tr}(PP^{-1}M)=\operatorname{tr}M$. So it is independent of choice of basis.
(ii) $∀S,S',T,T'∈\operatorname{Hom}(V, V),\ f_S(T+λT')=\operatorname{tr}\big(S∘(T+λT')\big)=\operatorname{tr}(S∘T+λS∘T')=\operatorname{tr}(S∘T)+λ\operatorname{tr}(S∘T')=f_S(T)+λf_S(T')⇒f_S$ is linear.
$f_{S+λS'}(T)=\operatorname{tr}\big((S+λS')∘T\big)=\operatorname{tr}(S∘T+λS'∘T)=\operatorname{tr}(S∘T)+λ\operatorname{tr}(S'∘T)=(f_S+λf_{S'})(T)⇒f_{S+λS'}=f_S+λf_{S'}⇒$the map $S ↦ f_S$ is linear.
Suppose $f_S=0$. Let $M$ be the matrix for $S$. Let $E_{ji}$ be the matrix (of same dimension as $M$) whose only non-zero entry is 1 at position $(j,i)$. $∀i,j:M_{ij}=\operatorname{tr}(ME_{ji})=0⇒M=0⇒S=0$. Therefore $S↦f_S$ is injective. Since the dimension of $\operatorname{Hom}(V, V)$ is equal to the dimension of its dual, $S↦f_S$ is an isomorphism.
By (i), $S↦f_S$ does not depend on a choice of basis. -
Let $T: V → W$ be a map between finite-dimensional vector spaces and let $T'': V'' → W''$ be the associated map between double duals.
Show that under the natural identifications between spaces and their double duals, $T''$ is identified with $T$. That is, if $E^{(V)}:V≅V''$ and $E^{(W)}:W≅W''$ are the natural isomorphisms, then we have
$$
T''∘E^{(V)}=E^{(W)}∘T.
$$
Proof.
$∀v∈V,f∈V',\ (T''∘E^{(V)})(v)(f)=E^{(V)}(v)\big(T'(f)\big)=(f∘T)(v)$
$(E^{(W)}∘T)(v)(f)=E^{(W)}\big(T(v)\big)(f)=(f∘T)(v)$
Therefore $T''∘E^{(V)}=E^{(W)}∘T$. - Let $V$ be finite dimensional. A hyperplane in $V$ is defined as the kernel of a linear functional.
Show that every subspace of $V$ is the intersection of hyperplanes.
Proof.
Let $W$ be a subspace of $V$. $∀x∈V∖W,∃Φ(x)∈V'$ such that $Φ(x)|_W= 0$ and $Φ(x)(x)=1$. Then $W = \bigcap_{x∉W} \ker Φ(x)$.
Show that every subspace of $V$ is the finite intersection of hyperplanes.
Proof.
Let ℬ be a basis of $W^0$. Then $W = \bigcap_{f∈ℬ} \ker f$.