Let $A$ be an $n\times n$ real symmetric matrix.
Prove that there exists an eigenvalue $\lambda$ of $A$ such that for any vector $\mathbf{v}\in \R^n$, we have the inequality
\[\mathbf{v}\cdot A\mathbf{v} \leq \lambda \|\mathbf{v}\|^2.\]
Add to solve later Determine whether each of the following statements is True or False.
(a) If $A$ and $B$ are $n \times n$ matrices, and $P$ is an invertible $n \times n$ matrix such that $A=PBP^{-1}$, then $\det(A)=\det(B)$.
(b) If the characteristic polynomial of an $n \times n$ matrix $A$ is
\[p(\lambda)=(\lambda-1)^n+2,\]
then $A$ is invertible.
(c) If $A^2$ is an invertible $n\times n$ matrix, then $A^3$ is also invertible.
(d) If $A$ is a $3\times 3$ matrix such that $\det(A)=7$, then $\det(2A^{\trans}A^{-1})=2$.
(e) If $\mathbf{v}$ is an eigenvector of an $n \times n$ matrix $A$ with corresponding eigenvalue $\lambda_1$, and if $\mathbf{w}$ is an eigenvector of $A$ with corresponding eigenvalue $\lambda_2$, then $\mathbf{v}+\mathbf{w}$ is an eigenvector of $A$ with corresponding eigenvalue $\lambda_1+\lambda_2$.
(Stanford University, Linear Algebra Exam Problem)
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Add to solve later Let $A$ be an $n\times n$ idempotent matrix, that is, $A^2=A$. Then prove that $A$ is diagonalizable.
Add to solve later Let $A$ and $B$ be $n\times n$ matrices.
Suppose that $A$ and $B$ have the same eigenvalues $\lambda_1, \dots, \lambda_n$ with the same corresponding eigenvectors $\mathbf{x}_1, \dots, \mathbf{x}_n$.
Prove that if the eigenvectors $\mathbf{x}_1, \dots, \mathbf{x}_n$ are linearly independent, then $A=B$.
Add to solve later Determine all $2\times 2$ matrices $A$ such that $A$ has eigenvalues $2$ and $-1$ with corresponding eigenvectors
\[\begin{bmatrix}
1 \\
0
\end{bmatrix} \text{ and } \begin{bmatrix}
2 \\
1
\end{bmatrix},\]
respectively.
Add to solve later (a) Let $A$ be a real orthogonal $n\times n$ matrix. Prove that the length (magnitude) of each eigenvalue of $A$ is $1$.
(b) Let $A$ be a real orthogonal $3\times 3$ matrix and suppose that the determinant of $A$ is $1$. Then prove that $A$ has $1$ as an eigenvalue.
Add to solve later Let $n$ be an odd integer and let $A$ be an $n\times n$ real matrix.
Prove that the matrix $A$ has at least one real eigenvalue.
Add to solve later Let $A$ be an $n\times n$ matrix. Suppose that $\mathbf{y}$ is a nonzero row vector such that
\[\mathbf{y}A=\mathbf{y}.\]
(Here a row vector means a $1\times n$ matrix.)
Prove that there is a nonzero column vector $\mathbf{x}$ such that
\[A\mathbf{x}=\mathbf{x}.\]
(Here a column vector means an $n \times 1$ matrix.)
Add to solve later Let $A$ be an $n\times n$ real matrix.
Prove that if $\lambda$ is an eigenvalue of $A$, then its complex conjugate $\bar{\lambda}$ is also an eigenvalue of $A$.
Add to solve later Let $A$ be an $n\times n$ matrix. Suppose that $A$ has real eigenvalues $\lambda_1, \lambda_2, \dots, \lambda_n$ with corresponding eigenvectors $\mathbf{u}_1, \mathbf{u}_2, \dots, \mathbf{u}_n$.
Furthermore, suppose that
\[|\lambda_1| > |\lambda_2| \geq \cdots \geq |\lambda_n|.\]
Let
\[\mathbf{x}_0=c_1\mathbf{u}_1+c_2\mathbf{u}_2+\cdots+c_n\mathbf{u}_n\]
for some real numbers $c_1, c_2, \dots, c_n$ and $c_1\neq 0$.
Define
\[\mathbf{x}_{k+1}=A\mathbf{x}_k \text{ for } k=0, 1, 2,\dots\]
and let
\[\beta_k=\frac{\mathbf{x}_k\cdot \mathbf{x}_{k+1}}{\mathbf{x}_k \cdot \mathbf{x}_k}=\frac{\mathbf{x}_k^{\trans} \mathbf{x}_{k+1}}{\mathbf{x}_k^{\trans} \mathbf{x}_k}.\]
Prove that
\[\lim_{k\to \infty} \beta_k=\lambda_1.\]
Add to solve later Find all the eigenvalues and eigenvectors of the matrix
\[A=\begin{bmatrix}
10001 & 3 & 5 & 7 &9 & 11 \\
1 & 10003 & 5 & 7 & 9 & 11 \\
1 & 3 & 10005 & 7 & 9 & 11 \\
1 & 3 & 5 & 10007 & 9 & 11 \\
1 &3 & 5 & 7 & 10009 & 11 \\
1 &3 & 5 & 7 & 9 & 10011
\end{bmatrix}.\]
(MIT, Linear Algebra Homework Problem)
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Add to solve later Find all eigenvalues of the matrix
\[A=\begin{bmatrix}
0 & i & i & i \\
i &0 & i & i \\
i & i & 0 & i \\
i & i & i & 0
\end{bmatrix},\]
where $i=\sqrt{-1}$. For each eigenvalue of $A$, determine its algebraic multiplicity and geometric multiplicity.
Add to solve later Let
\[A=\begin{bmatrix}
2 & -1 & -1 \\
-1 &2 &-1 \\
-1 & -1 & 2
\end{bmatrix}.\]
Determine whether the matrix $A$ is diagonalizable. If it is diagonalizable, then diagonalize $A$.
That is, find a nonsingular matrix $S$ and a diagonal matrix $D$ such that $S^{-1}AS=D$.
Add to solve later Consider the matrix
\[A=\begin{bmatrix}
3/2 & 2\\
-1& -3/2
\end{bmatrix} \in M_{2\times 2}(\R).\]
(a) Find the eigenvalues and corresponding eigenvectors of $A$.
(b) Show that for $\mathbf{v}=\begin{bmatrix}
1 \\
0
\end{bmatrix}\in \R^2$, we can choose $n$ large enough so that the length $\|A^n\mathbf{v}\|$ is as small as we like.
(University of California, Berkeley, Linear Algebra Final Exam Problem)
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Add to solve later Find the determinant of the following matrix
\[A=\begin{bmatrix}
6 & 2 & 2 & 2 &2 \\
2 & 6 & 2 & 2 & 2 \\
2 & 2 & 6 & 2 & 2 \\
2 & 2 & 2 & 6 & 2 \\
2 & 2 & 2 & 2 & 6
\end{bmatrix}.\]
(Harvard University, Linear Algebra Exam Problem)
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Add to solve later Find all the eigenvalues and eigenvectors of the matrix
\[A=\begin{bmatrix}
3 & 9 & 9 & 9 \\
9 &3 & 9 & 9 \\
9 & 9 & 3 & 9 \\
9 & 9 & 9 & 3
\end{bmatrix}.\]
(Harvard University, Linear Algebra Final Exam Problem)
Add to solve later Let $A$ be an $n \times n$ matrix and let $c$ be a complex number.
(a) For each eigenvalue $\lambda$ of $A$, prove that $\lambda+c$ is an eigenvalue of the matrix $A+cI$, where $I$ is the identity matrix. What can you say about the eigenvectors corresponding to $\lambda+c$?
(b) Prove that the algebraic multiplicity of the eigenvalue $\lambda$ of $A$ is the same as the algebraic multiplicity of the eigenvalue $\lambda+c$ of $A+cI$ are equal.
(c) How about geometric multiplicities?
Add to solve later Let $A$ be an $n\times n$ idempotent complex matrix.
Then prove that $A$ is diagonalizable.
Add to solve later (a) Let
\[A=\begin{bmatrix}
0 & 0 & 0 & 0 \\
1 &1 & 1 & 1 \\
0 & 0 & 0 & 0 \\
1 & 1 & 1 & 1
\end{bmatrix}.\]
Find the eigenvalues of the matrix $A$. Also give the algebraic multiplicity of each eigenvalue.
(b) Let
\[A=\begin{bmatrix}
0 & 0 & 0 & 0 \\
1 &1 & 1 & 1 \\
0 & 0 & 0 & 0 \\
1 & 1 & 1 & 1
\end{bmatrix}.\]
One of the eigenvalues of the matrix $A$ is $\lambda=0$. Find the geometric multiplicity of the eigenvalue $\lambda=0$.
Add to solve later 
