# Is inner product preserved on change of basis?

Suppose we have a matrix $$A$$ over which an inner product is defined. Let us denote this inner product by $$\langle, \rangle_A$$.

Now let us suppose that this matrix $$A$$ is symmetric. Then there exists a matrix $$Q$$ such that $$Q^TAQ$$ is a diagonal matrix. For any two vectors $$v$$ and $$w$$, is $$\langle v,w\rangle_A=\langle v,w\rangle_{Q^TAQ}$$?

In other words, is inner product preserved when we change the basis of $$R^n$$?

• The basis has to be orthogonal. – GNUSupporter 8964民主女神 地下教會 Feb 26 at 20:33
• @GNUSupporter8964民主女神地下教會- That's not necessarily true when diagonalizing a symmetric matrix though, right? – Anju George Feb 26 at 20:35
• It's a well known result in linear algebra that symmetric matrices iff orthogonal eigenbasis. I'm simply addressing the last prompt alone without previous context. – GNUSupporter 8964民主女神 地下教會 Feb 26 at 20:39
• Is the inner product defined over $A$, or the inner product defined using $A$ over $\mathbb R^n$ using a canonical basis? – Sandesh Jr Mar 1 at 10:32