Eigenvalues and Eigenvectors

This matrix has two eigenvalues $\lambda_{1}=3$, $\lambda_{2}=-4$.

Give two different diagonal form matrices with which it is similar.

For each, find the characteristic polynomial and the eigenvalues.

• (a)  $\begin{pmatrix}10&-9\\ 4&-2\end{pmatrix}$     (b)  $\begin{pmatrix}1&2\\ 4&3\end{pmatrix}$     (c)  $\begin{pmatrix}0&3\\ 7&0\end{pmatrix}$     (d)  $\begin{pmatrix}0&0\\ 0&0\end{pmatrix}$     (e)  $\begin{pmatrix}1&0\\ 0&1\end{pmatrix}$

For each matrix, find the characteristic equation and the eigenvalues and associated eigenvectors.

• (a)  $\begin{pmatrix}3&0\\ 8&-1\end{pmatrix}$     (b)  $\begin{pmatrix}3&2\\ -1&0\end{pmatrix}$

Find the characteristic equation, and the eigenvalues and associated eigenvectors for this matrix. Hint. The eigenvalues are complex.

Find the characteristic polynomial, the eigenvalues, and the associated eigenvectors of this matrix.

For each matrix, find the characteristic equation, and the eigenvalues and associated eigenvectors.

• (a)  $\begin{pmatrix}3&-2&0\\ -2&3&0\\ 0&0&5\end{pmatrix}$     (b)  $\begin{pmatrix}0&1&0\\ 0&0&1\\ 4&-17&8\end{pmatrix}$

For each matrix, find the characteristic polynomial, and the eigenvalues and associated eigenspaces. Also find the algebraic and geometric multiplicities.

• (a)  $\begin{pmatrix}13&-4\\ -4&7\end{pmatrix}$     (b)  $\begin{pmatrix}1&3&-3\\ -3&7&-3\\ -6&6&-2\end{pmatrix}$     (c)  $\begin{pmatrix}2&3&-3\\ 0&2&-3\\ 0&0&1\end{pmatrix}$

Let $t\colon\mathcal{P}_{2}\to\mathcal{P}_{2}$ be this linear map.

Find its eigenvalues and the associated eigenvectors.

Find the eigenvalues and eigenvectors of this map $t\colon\mathcal{M}_{2}\to\mathcal{M}_{2}$.

Find the eigenvalues and associated eigenvectors of the differentiation operator $d/dx\colon\mathcal{P}_{3}\to\mathcal{P}_{3}$.

Prove that the eigenvalues of a triangular matrix (upper or lower triangular) are the entries on the diagonal.

This matrix has distinct eigenvalues.

• Diagonalize it.

• Find a basis with respect to which this matrix has that diagonal representation.

• Draw the diagram. Find the matrices $P$ and $P^{-1}$ to effect the change of basis.

Find the formula for the characteristic polynomial of a $2\!\times\!2$ matrix.

Prove that the characteristic polynomial of a transformation is well-defined.

Prove or disprove: if all the eigenvalues of a matrix are $0$ then it must be the zero matrix.

• Show that any non-$\vec{0}$ vector in any nontrivial vector space can be an eigenvector. That is, given a $\vec{v}\neq\vec{0}$ from a nontrivial $V$, show that there is a transformation $t\colon V\to V$ having a scalar eigenvalue $\lambda\in\mathbb{R}$ such that $\vec{v}\in V_{\lambda}$.

• What if we are given a scalar $\lambda$? Can any non-$\vec{0}$ member of any nontrivial vector space be an eigenvector associated with $\lambda$?

Suppose that $t\colon V\to V$ and $T={\rm Rep}_{B,B}(t)$. Prove that the eigenvectors of $T$ associated with $\lambda$ are the non-$\vec{0}$ vectors in the kernel of the map represented (with respect to the same bases) by $T-\lambda I$.

Prove that if $a,\ldots,\,d$ are all integers and $a+b=c+d$ then

has integral eigenvalues, namely $a+b$ and $a-c$.

Prove that if $T$ is nonsingular and has eigenvalues $\lambda_{1},\dots,\lambda_{n}$ then $T^{-1}$ has eigenvalues $1/\lambda_{1},\dots,1/\lambda_{n}$. Is the converse true?

Suppose that $T$ is $n\!\times\!n$ and $c,d$ are scalars.

• Prove that if $T$ has the eigenvalue $\lambda$ with an associated eigenvector $\vec{v}$ then $\vec{v}$ is an eigenvector of $cT+dI$ associated with eigenvalue $c\lambda+d$.

• Prove that if $T$ is diagonalizable then so is $cT+dI$.

Show that $\lambda$ is an eigenvalue of $T$ if and only if the map represented by $T-\lambda I$ is not an isomorphism.

• Show that if $\lambda$ is an eigenvalue of $A$ then $\lambda^{k}$ is an eigenvalue of $A^{k}$.

• What is wrong with this proof generalizing that? “If $\lambda$ is an eigenvalue of $A$ and $\mu$ is an eigenvalue for $B$, then $\lambda\mu$ is an eigenvalue for $AB$, for, if $A\vec{x}=\lambda\vec{x}$ and $B\vec{x}=\mu\vec{x}$ then $AB\vec{x}=A\mu\vec{x}=\mu A\vec{x}=\mu\lambda\vec{x}$”?

Do matrix equivalent matrices have the same eigenvalues?

Show that a square matrix with real entries and an odd number of rows has at least one real eigenvalue.

Diagonalize.

Suppose that $P$ is a nonsingular $n\!\times\!n$ matrix. Show that the similarity transformation map $t_{P}\colon\mathcal{M}_{n\!\times\!n}\to\mathcal{M}_{n\!\times\!n}$ sending $T\mapsto PTP^{-1}$ is an isomorphism.

Show that if $A$ is an $n$ square matrix and each row (column) sums to $c$ then $c$ is a characteristic root of $A$. (“Characteristic root” is a synonym for eigenvalue.)