## A Symmetric Positive Definite Matrix and An Inner Product on a Vector Space

## 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.

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