# Tagged: positive definite matrix

## Problem 559

For each of the following matrix $A$, prove that $\mathbf{x}^{\trans}A\mathbf{x} \geq 0$ for all vectors $\mathbf{x}$ in $\R^2$. Also, determine those vectors $\mathbf{x}\in \R^2$ such that $\mathbf{x}^{\trans}A\mathbf{x}=0$.

(a) $A=\begin{bmatrix} 4 & 2\\ 2& 1 \end{bmatrix}$.

(b) $A=\begin{bmatrix} 2 & 1\\ 1& 3 \end{bmatrix}$.

## Problem 539

Consider the $2\times 2$ real matrix
$A=\begin{bmatrix} 1 & 1\\ 1& 3 \end{bmatrix}.$

(a) Prove that the matrix $A$ is positive definite.

(b) Since $A$ is positive definite by part (a), the formula
$\langle \mathbf{x}, \mathbf{y}\rangle:=\mathbf{x}^{\trans} A \mathbf{y}$ for $\mathbf{x}, \mathbf{y} \in \R^2$ defines an inner product on $\R^n$.
Consider $\R^2$ as an inner product space with this inner product.

Prove that the unit vectors
$\mathbf{e}_1=\begin{bmatrix} 1 \\ 0 \end{bmatrix} \text{ and } \mathbf{e}_2=\begin{bmatrix} 0 \\ 1 \end{bmatrix}$ are not orthogonal in the inner product space $\R^2$.

(c) Find an orthogonal basis $\{\mathbf{v}_1, \mathbf{v}_2\}$ of $\R^2$ from the basis $\{\mathbf{e}_1, \mathbf{e}_2\}$ using the Gram-Schmidt orthogonalization process.

## Problem 538

(a) Suppose that $A$ is an $n\times n$ real symmetric positive definite matrix.
Prove that
$\langle \mathbf{x}, \mathbf{y}\rangle:=\mathbf{x}^{\trans}A\mathbf{y}$ defines an inner product on the vector space $\R^n$.

(b) Let $A$ be an $n\times n$ real matrix. Suppose that
$\langle \mathbf{x}, \mathbf{y}\rangle:=\mathbf{x}^{\trans}A\mathbf{y}$ defines an inner product on the vector space $\R^n$.

Prove that $A$ is symmetric and positive definite.

## Problem 514

Prove that a positive definite matrix has a unique positive definite square root.