Let A be s symmetric and positive definite $n\times n $ matrix show that $\|x\|_A=\sqrt {x^TAx}$ is a norm for vectors $x\in \mathbb{R^n}$
I need to prove this by the eigenvalues and eigenvector decomposition of symmetric positive definite matrices.
Suppose A be s symmetric and positive definite $n\times n $ matrix
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linear-algebra
inner-product-space
1 Answers
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Suppose $A = U \Lambda U^T $, where $U$ is orthogonal and $\Lambda$ is a diagonal matrix with positive entries on the diagonal. Then $\|x\|_A = \| \sqrt{\Lambda} U^T x\|_2$, and since $\sqrt{\Lambda} U^T$ is invertible, it follows that $\|\cdot\|_A$ is a norm.