7.E: Exercises for Chapter 7

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Contributed by Isaiah Lankham, Bruno Nachtergaele, & Anne Schilling
Professor (Mathematics) at University of California, Davis

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Calculational Exercises

1. Let $T \in \cal{L}(\mathbb{F}^2 , \mathbb{F}^2)$ be deﬁned by

\[T (u, v) = (v, u)\]

for every $u, v \in \mathbb{F}.$ Compute the eigenvalues and associated eigenvectors for $T.$

2. Let $T \in \cal{L}(\mathbb{F}^3 , \mathbb{F}^3$) be deﬁned by

\[T (u, v, w) = (2v, 0, 5w)\]

for every $u, v, w \in \mathbb{F}.$ Compute the eigenvalues and associated eigenvectors for $T.$

3. Let $n \in \mathbb{Z}_+$ be a positive integer and $T \in \cal{L}(\mathbb{F}^n , \mathbb{F}^n )$ be deﬁned by

\[T (x_1 ,\ldots, x_n ) = (x1 + \cdots + x_n , \ldots, x_1 + \cdots + x_n)\]

for every $x_1 ,\ldots, x_n \in \mathbb{F}.$ Compute the eigenvalues and associated eigenvectors for $T.$

4. Find eigenvalues and associated eigenvectors for the linear operators on $\mathbb{F}^2$ deﬁned by each given $2 \times 2$ matrix.

$(a) \left[ \begin{array}{cc} 3 & 0 \\ 8 & -1 \end{array} \right], ~~ (b) \left[ \begin{array}{cc} 10 & -9 \\ 4 & -2 \end{array} \right], ~~ (c) \left[ \begin{array}{cc} 0 & 3 \\ 4 & 0 \end{array} \right], $

$(d) \left[ \begin{array}{cc} -2 & -7 \\ 1 & 2 \end{array} \right], ~~ (e) \left[ \begin{array}{cc} 0 & 0 \\ 0 & 0 \end{array} \right], ~~ (f) \left[ \begin{array}{cc} 1 & 0 \\ 0 & 1 \end{array} \right] $

Hint: Use the fact that, given a matrix $A = \left[ \begin{array}{cc} a & b \\ c & d \end{array} \right] \in \mathbb{F}^{2\times2} , \lambda \in \mathbb{F}$ is an eigenvalue for A if and only if $(a − \lambda)(d − \lambda) − bc = 0.$

5. For each matrix $A$ below, ﬁnd eigenvalues for the induced linear operator $T$ on $\mathbb{F}^n$ without performing any calculations. Then describe the eigenvectors $v \in \mathbb{F}^n$ associated to each eigenvalue $\lambda$ by looking at solutions to the matrix equation $(A − \lambda I)v = 0,$ where I denotes the identity map on $\mathbb{F}^n.$

$(a) \left[ \begin{array}{cc} -1 & 6 \\ 0 & 5 \end{array} \right], ~~(b) \left[ \begin{array}{cccc} -\frac{1}{3} & 0 & 0 & 0 \\ 0 & -\frac{1}{3} & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & \frac{1}{2} \end{array} \right], ~~(c) \left[ \begin{array}{cccc} 1 & 3 & 7 & 11 \\ 0 & \frac{1}{2} & 3 & 8 \\ 0 & 0 & 0 & 4 \\ 0 & 0 & 0 & 2 \end{array} \right]$

6. For each matrix $A$ below, describe the invariant subspaces for the induced linear operator $T on \mathbb{F}^2$ that maps each $v \in \mathbb{F}^2$ to $T (v) = Av.$

$(a) \left[ \begin{array}{cc} 4 & -1 \\ 2 & 1 \end{array} \right], ~~ (b) \left[ \begin{array}{cc} 0 & 1 \\ -1 & 0 \end{array} \right], ~~ (c) \left[ \begin{array}{cc} 2 & 3 \\ 0 & 2 \end{array} \right], ~~ (d) \left[ \begin{array}{cc} 1 & 0 \\ 0 & 0 \end{array} \right] $

7. Let $T \in \cal{L}(\mathbb{R}^2)$ be deﬁned by

\[ T \left( \begin{array}{c} x \\ y \end{array} \right) = \left( \begin{array}{c} x \\ x+y \end{array} \right),~~\rm{~for ~all~} \left( \begin{array}{c} x \\ y \end{array} \right) \in \mathbb{R}^2.\]

Deﬁne two real numbers $\lambda_+$ and $\lambda_−$ as follows:

\[ \lambda_+=\frac{1+\sqrt{5}}{2}, ~~ \lambda_-=\frac{1-\sqrt{5}}{2}.\]

(a) Find the matrix of $T$ with respect to the canonical basis for $\mathbb{R}^2$ (both as the domain and the codomain of $T$ ; call this matrix $A$).
(b) Verify that $\lambda_+$ and $\lambda_−$ are eigenvalues of $T$ by showing that $v_+$ and $v_−$ are eigen-
vectors, where

\[ v_+ = \left( \begin{array}{c} 1 \\ \lambda_+ \end{array} \right),~~ v_- = \left( \begin{array}{c} 1 \\ \lambda_- \end{array} \right).\]

(c) Show that $(v_+ , v_− )$ is a basis of $\mathbb{R}^2.$
(d) Find the matrix of $T$ with respect to the basis $(v_+ , v_− )$ for $\mathbb{R}^2$ (both as the domain
and the codomain of $T$ ; call this matrix $B$).

Proof-Writing Exercises

1. Let $V$ be a ﬁnite-dimensional vector space over $\mathbb{F}$ with $T \in \cal{L}$$(V, V),$ and let $U_1 , \ldots, U_m$ be subspaces of $V$ that are invariant under $T $. Prove that $U_1 + \cdots + U_m$ must then also be an invariant subspace of $V$ under $T .$

2. Let $V$ be a ﬁnite-dimensional vector space over $\mathbb{F }$ with $T \in \cal{L}$$(V, V ),$ and suppose that $U_1$ and $U_2$ are subspaces of $V$ that are invariant under $T $. Prove that $U_1 \cap U_2$ is also an invariant subspace of $V$ under $T.$

3. Let $V$ be a ﬁnite-dimensional vector space over $\mathbb{F}$ with $T \in \cal{L}$$(V, V)$ invertible and $\lambda \in \mathbb{F} = \{0\}.$ Prove $\lambda$ is an eigenvalue for $T$ if and only if $\lambda−1$ is an eigenvalue for $T −1.$

4. Let $V$ be a ﬁnite-dimensional vector space over $\mathbb{F},$ and suppose that $T \in \cal{L}$$(V, V)$ has the property that every $v \in V$ is an eigenvector for $T$ . Prove that $T$ must then be a scalar multiple of the identity function on $V.$

5. Let $V$ be a ﬁnite-dimensional vector space over $\mathbb{F},$ and let $S, T \in \cal{L}$$(V)$ be linear operators on $V$ with $S$ invertible. Given any polynomial $p(z) \in \mathbb{F}[z],$ prove that

\[p(S \circ T \circ S^{ −1} ) = S \circ p(T ) \circ S^{ −1}.\]

6. Let $V$ be a ﬁnite-dimensional vector space over $\mathbb{C}, T \in \cal{L}$$(V)$ be a linear operator on $V$ , and $p(z) \in \mathbb{C}[z]$ be a polynomial. Prove that $\lambda \in \mathbb{C}$ is an eigenvalue of the linear operator $p(T ) \in \cal{L}$$(V)$ if and only if $T$ has an eigenvalue $\mu \in \mathbb{C}$ such that $p(\mu) = \lambda.$

7. Let $V$ be a ﬁnite-dimensional vector space over $\mathbb{C}$ with $T \in \cal{L}$$(V)$ a linear operator on $V.$ Prove that, for each $k = 1,\ldots , dim(V ),$ there is an invariant subspace $U_k$ of $V$ under $T$ such that $dim(U_k ) = k.$

8. Prove or give a counterexample to the following claim:

Claim. Let $V$ be a ﬁnite-dimensional vector space over $\mathbb{F},$ and let $T \in \cal{L}$$(V)$ be a linear operator on $V$ . If the matrix for $T$ with respect to some basis on $V$ has all zeros on the diagonal, then $T$ is not invertible.

9. Prove or give a counterexample to the following claim:

Claim. Let $V$ be a ﬁnite-dimensional vector space over $\mathbb{F},$ and let $T \in \cal{L}$$(V)$ be a linear
operator on $V$ . If the matrix for $T$ with respect to some basis on $V$ has all non-zero elements on the diagonal, then $T$ is invertible.

10. Let $V$ be a ﬁnite-dimensional vector space over $\mathbb{F},$ and let $S, T \in \cal{L}$$(V)$ be linear operators on $V$ . Suppose that $T$ has $dim(V)$ distinct eigenvalues and that, given any eigenvector $v \in V$ for $T$ associated to some eigenvalue $\lambda \in \mathbb{F},$ $v$ is also an eigenvector
for $S$ associated to some (possibly distinct) eigenvalue $\mu \in \mathbb{F}.$ Prove that $T \circ S = S \circ T .$

11. Let $V$ be a ﬁnite-dimensional vector space over $\mathbb{F},$ and suppose that the linear operator $P \in \cal{L}$$(V)$ has the property that $P^2 = P.$ Prove that $V = null(P ) \oplus range(P ).$

12. (a) Let $a, b, c, d \in \mathbb{F}$ and consider the system of equations given by

\[ax_1 + bx_2 = 0 \\ cx_1 + dx_2 = 0.\]

Note that $x_1 = x_2 = 0$ is a solution for any choice of $a, b, c,$ and $d$. Prove that this system of equations has a non-trivial solution if and only if $ad − bc = 0.$

(b) Let $A = \left[ \begin{array}{cc} a & b \\ c & d \end{array} \right] \in \mathbb{F}^{2\times2}$ , and recall that we can deﬁne a linear operator $T \in \cal{L}(\mathbb{F}^2 )$ on $\mathbb{F}^2$ by setting $T (v) = Av$ for each $v = \left[ \begin{array}{c} v_1 \\ v_2 \end{array} \right] \in \mathbb{F}^2. $

Show that the eigenvalues for $T$ are exactly the $\lambda \in \mathbb{F}$ for which $p(\lambda) = 0,$ where $p(z) = (a − z)(d − z) − bc.$

Hint: Write the eigenvalue equation $Av = \lambda v$ as $(A − \lambda I)v = 0$ and use the ﬁrst part.

Contributors

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