What does it mean for a matrix to induce a norm?
I'm given the following math problem: Consider the scalar product $\langle x,y\rangle = x^TAy$ given by the matrix:
$$ A= \begin{pmatrix} 1 & 0 & 2 \\ 0 & 2 & 0 \\ 2 & 0 & 8 \\ \end{pmatrix} $$
Determine the length of $(2, 1, 0)^T$ with respect to the norm induced by the scalar product.
What does it mean for a norm to be induced by something? In the answers they say that the norm is induced by the matrix $A$. Does it mean that this vector is not in a normal coordinate system but somehow inside this matrix?
Thanks for the help!
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$\begingroup$A matrix $A$ which is symmetric and positive definite defines (as your posts states) an inner product: $$ \langle x, y\rangle \stackrel{\rm def}{=} x^T A y, \tag{$\forall x,y$} $$
Now, an inner product defines (induces) a norm by setting $\lVert x\rVert \stackrel{\rm def}{=} \sqrt{\langle x, x\rangle}$.
So "a symmetric positive definite matrix $A$ induces a norm."
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