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
\[\mathbf{v}\cdot A\mathbf{v} \leq \lambda \|\mathbf{v}\|^2.\]

Since these eigenvalues are real numbers, there is the largest one.
Let $\lambda$ be the largest eigenvalue of $A$.
With this choice of $\lambda$ we show that the inequality
\[\mathbf{v}\cdot A\mathbf{v} \leq \lambda \|\mathbf{v}\|^2\]
holds for any $\mathbf{v}\in \R^n$.

Also recall that for a real symmetric matrix, there are eigenvalues $\mathbf{v}_1, \dots, \mathbf{v}_n$ corresponding to $\lambda_1, \dots, \lambda_n$ such that
\[B=\{\mathbf{v}_1, \dots, \mathbf{v}_n\}\]
form an orthonormal basis of $\R^n$.
(This statement is equivalent to that every real symmetric matrix is diagonalizable by an orthogonal matrix.)

Let $\mathbf{v}$ be an arbitrary vector in $\R^n$.
Then since $B$ is a basis of $\R^n$, we can write
\[\mathbf{v}=c_1\mathbf{v}_1+\dots+c_n\mathbf{v}_n\]
for some $c_1, \dots, c_n\in \R$.

Then we calculate
\begin{align*}
A\mathbf{v}&=A(c_1\mathbf{v}_1+\dots+c_n\mathbf{v}_n)\\
&=c_1A\mathbf{v}_1+\dots+c_nA\mathbf{v}_n\\
&=c_1\lambda_1\mathbf{v}_1+\dots+c_n\lambda_n\mathbf{v}_n
\end{align*}
since $A\mathbf{v}_i=\lambda_i\mathbf{v}_i$ for $i=1, \dots, n$.

Using this, we have
\begin{align*}
\mathbf{v}\cdot A\mathbf{v}&=(c_1\mathbf{v}_1+\dots+c_n\mathbf{v}_n)\cdot (c_1\lambda_1\mathbf{v}_1+\dots+c_n\lambda_n\mathbf{v}_n)\\
&=c_1^2\lambda_1+\cdots+c_n^2\lambda_n.
\end{align*}

Here, we used that $B=\{\mathbf{v}_1, \dots, \mathbf{v}_n\}$ is an orthonormal basis of $\R^3$.
That is, we used the properties
\begin{align*}
\mathbf{v}_i\cdot \mathbf{v}_j=\begin{cases}
1 & \text{if } i=j\\
0 & \text{if } i\neq j.
\end{cases}
\end{align*}

Since $\lambda$ is the largest eigenvalue of $A$, we have
\begin{align*}
\mathbf{v}\cdot A\mathbf{v}&=c_1^2\lambda_1+\cdots+c_n^2\lambda_n\\
& \leq c_1^2\lambda+\cdots+c_n^2\lambda\\
&=\lambda(c_2^2+\cdots+c_n^2)\\
&=\lambda \|\mathbf{v}\|^2.
\end{align*}
Hence the required inequality holds.

A Matrix Equation of a Symmetric Matrix and the Limit of its Solution
Let $A$ be a real symmetric $n\times n$ matrix with $0$ as a simple eigenvalue (that is, the algebraic multiplicity of the eigenvalue $0$ is $1$), and let us fix a vector $\mathbf{v}\in \R^n$.
(a) Prove that for sufficiently small positive real $\epsilon$, the equation […]

Diagonalizable by an Orthogonal Matrix Implies a Symmetric Matrix
Let $A$ be an $n\times n$ matrix with real number entries.
Show that if $A$ is diagonalizable by an orthogonal matrix, then $A$ is a symmetric matrix.
Proof.
Suppose that the matrix $A$ is diagonalizable by an orthogonal matrix $Q$.
The orthogonality of the […]

Quiz 13 (Part 1) Diagonalize a Matrix
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 […]

Maximize the Dimension of the Null Space of $A-aI$
Let
\[ A=\begin{bmatrix}
5 & 2 & -1 \\
2 &2 &2 \\
-1 & 2 & 5
\end{bmatrix}.\]
Pick your favorite number $a$. Find the dimension of the null space of the matrix $A-aI$, where $I$ is the $3\times 3$ identity matrix.
Your score of this problem is equal to that […]

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Answer the following two questions with justification.
(a) Does there exist a $2 \times 2$ matrix $A$ with $A^3=O$ but $A^2 \neq O$? Here $O$ denotes the $2 \times 2$ zero matrix.
(b) Does there exist a $3 \times 3$ real matrix $B$ such that $B^2=A$ […]

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Let $A$ be an $n\times n$ real symmetric matrix whose eigenvalues are all non-negative real numbers.
Show that there is an $n \times n$ real matrix $B$ such that $B^2=A$.
Hint.
Use the fact that a real symmetric matrix is diagonalizable by a real orthogonal matrix.
[…]

Diagonalize the $2\times 2$ Hermitian Matrix by a Unitary Matrix
Consider the Hermitian matrix
\[A=\begin{bmatrix}
1 & i\\
-i& 1
\end{bmatrix}.\]
(a) Find the eigenvalues of $A$.
(b) For each eigenvalue of $A$, find the eigenvectors.
(c) Diagonalize the Hermitian matrix $A$ by a unitary matrix. Namely, find a diagonal matrix […]

Let $\mathbf{u}=\begin{bmatrix} 1 \\ 1 \\ 0 \end{bmatrix}$ and $T:\R^3 \to \R^3$ be the linear transformation \[T(\mathbf{x})=\proj_{\mathbf{u}}\mathbf{x}=\left(\, \frac{\mathbf{u}\cdot \mathbf{x}}{\mathbf{u}\cdot \mathbf{u}}...

## 1 Response

[…] For a proof of this problem, see the post “Inequality about Eigenvalue of a Real Symmetric Matrix“. […]