Sequence Converges to the Largest Eigenvalue of a Matrix
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 […]

Determine a Condition on $a, b$ so that Vectors are Linearly Dependent
Let
\[\mathbf{v}_1=\begin{bmatrix}
1 \\
2 \\
0
\end{bmatrix}, \mathbf{v}_2=\begin{bmatrix}
1 \\
a \\
5
\end{bmatrix}, \mathbf{v}_3=\begin{bmatrix}
0 \\
4 \\
b
\end{bmatrix}\]
be vectors in $\R^3$.
Determine a […]

Diagonalizable Matrix with Eigenvalue 1, -1
Suppose that $A$ is a diagonalizable $n\times n$ matrix and has only $1$ and $-1$ as eigenvalues.
Show that $A^2=I_n$, where $I_n$ is the $n\times n$ identity matrix.
(Stanford University Linear Algebra Exam)
See below for a generalized problem.
Hint.
Diagonalize the […]

Rotation Matrix in the Plane and its Eigenvalues and Eigenvectors
Consider the $2\times 2$ matrix
\[A=\begin{bmatrix}
\cos \theta & -\sin \theta\\
\sin \theta& \cos \theta \end{bmatrix},\]
where $\theta$ is a real number $0\leq \theta < 2\pi$.
(a) Find the characteristic polynomial of the matrix $A$.
(b) Find the […]

Inequality Regarding Ranks of Matrices
Let $A$ be an $n \times n$ matrix over a field $K$. Prove that
\[\rk(A^2)-\rk(A^3)\leq \rk(A)-\rk(A^2),\]
where $\rk(B)$ denotes the rank of a matrix $B$.
(University of California, Berkeley, Qualifying Exam)
Hint.
Regard the matrix as a linear transformation $A: […]

Compute Power of Matrix If Eigenvalues and Eigenvectors Are Given
Let $A$ be a $3\times 3$ matrix. Suppose that $A$ has eigenvalues $2$ and $-1$, and suppose that $\mathbf{u}$ and $\mathbf{v}$ are eigenvectors corresponding to $2$ and $-1$, respectively, where
\[\mathbf{u}=\begin{bmatrix}
1 \\
0 \\
-1
\end{bmatrix} \text{ […]

Is the Set of All Orthogonal Matrices a Vector Space?
An $n\times n$ matrix $A$ is called orthogonal if $A^{\trans}A=I$.
Let $V$ be the vector space of all real $2\times 2$ matrices.
Consider the subset
\[W:=\{A\in V \mid \text{$A$ is an orthogonal matrix}\}.\]
Prove or disprove that $W$ is a subspace of […]