Suppose that a real symmetric matrix $A$ has two distinct eigenvalues $\alpha$ and $\beta$.
Show that any eigenvector corresponding to $\alpha$ is orthogonal to any eigenvector corresponding to $\beta$.

(Nagoya University, Linear Algebra Final Exam Problem)

Two vectors $\mathbf{u}$ and $\mathbf{v}$ are orthogonal if their inner (dot) product $\mathbf{u}\cdot \mathbf{v}:=\mathbf{u}^{\trans}\mathbf{v}=0$.

Here $\mathbf{u}^{\trans}$ is the transpose of $\mathbf{u}$.

A fact that we will use below is that for matrices $A$ and $B$, we have $(AB)^{\trans}=B^{\trans}A^{\trans}$.

Proof.

Let $\mathbf{u}, \mathbf{v}$ be eigenvectors corresponding to $\alpha, \beta$, respectively.
Namely we have
\[A\mathbf{u}=\alpha \mathbf{u} \text{ and } A\mathbf{v}=\beta \mathbf{v}. \tag{*}\]

To prove that $\mathbf{u}$ and $\mathbf{v}$ are orthogonal, we show that the inner product $\mathbf{u} \cdot \mathbf{v}=0$.
Keeping this in mind, we compute
\begin{align*}
&\alpha (\mathbf{u} \cdot \mathbf{v}) =(\alpha \mathbf{u}) \cdot \mathbf{v} \\
&\stackrel{(*)}{=} A\mathbf{u}\cdot \mathbf{v} =(A\mathbf{u})^{\trans} \mathbf{v}\\
&=\mathbf{u}^{\trans}A^{\trans}\mathbf{v} \text{ (This follows from the fact mentioned in the hint above)} \\
&=\mathbf{u}^{\trans}A\mathbf{v} \text{ (since $A$ is symmetric.)}\\
& \stackrel{(*)}{=} \mathbf{u}^{\trans}\beta \mathbf{v}=\beta (\mathbf{u}^{\trans} \mathbf{v})=\beta (\mathbf{u}\cdot \mathbf{v}).
\end{align*}

Therefore we obtain
\[\alpha (\mathbf{u} \cdot \mathbf{v})=\beta (\mathbf{u} \cdot \mathbf{v}),\]
and thus
\[(\alpha-\beta)(\mathbf{u} \cdot \mathbf{v})=0.\]

Since $\alpha$ and $\beta$ are distinct, $\alpha-\beta \neq 0$.
Hence we must have
\[\mathbf{u} \cdot \mathbf{v}=0,\]
and the eigenvectors $\mathbf{u}, \mathbf{v}$ are orthogonal.

Inner Product, Norm, and Orthogonal Vectors
Let $\mathbf{u}_1, \mathbf{u}_2, \mathbf{u}_3$ are vectors in $\R^n$. Suppose that vectors $\mathbf{u}_1$, $\mathbf{u}_2$ are orthogonal and the norm of $\mathbf{u}_2$ is $4$ and $\mathbf{u}_2^{\trans}\mathbf{u}_3=7$. Find the value of the real number $a$ in […]

Find the Eigenvalues and Eigenvectors of the Matrix $A^4-3A^3+3A^2-2A+8E$.
Let
\[A=\begin{bmatrix}
1 & -1\\
2& 3
\end{bmatrix}.\]
Find the eigenvalues and the eigenvectors of the matrix
\[B=A^4-3A^3+3A^2-2A+8E.\]
(Nagoya University Linear Algebra Exam Problem)
Hint.
Apply the Cayley-Hamilton theorem.
That is if $p_A(t)$ is the […]

Unit Vectors and Idempotent Matrices
A square matrix $A$ is called idempotent if $A^2=A$.
(a) Let $\mathbf{u}$ be a vector in $\R^n$ with length $1$.
Define the matrix $P$ to be $P=\mathbf{u}\mathbf{u}^{\trans}$.
Prove that $P$ is an idempotent matrix.
(b) Suppose that $\mathbf{u}$ and $\mathbf{v}$ be […]

Find the Limit of a Matrix
Let
\[A=\begin{bmatrix}
\frac{1}{7} & \frac{3}{7} & \frac{3}{7} \\
\frac{3}{7} &\frac{1}{7} &\frac{3}{7} \\
\frac{3}{7} & \frac{3}{7} & \frac{1}{7}
\end{bmatrix}\]
be $3 \times 3$ matrix. Find
\[\lim_{n \to \infty} A^n.\]
(Nagoya University Linear […]

Determine the Values of $a$ such that the 2 by 2 Matrix is Diagonalizable
Let
\[A=\begin{bmatrix}
1-a & a\\
-a& 1+a
\end{bmatrix}\]
be a $2\times 2$ matrix, where $a$ is a complex number.
Determine the values of $a$ such that the matrix $A$ is diagonalizable.
(Nagoya University, Linear Algebra Exam Problem)
Proof.
To […]

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 […]