# Playing with inner products

An inner product is defined as $(x,y)_E=[a_1\quad a_2 \quad\ldots \quad a_n]E^TE[b_1\quad b_2 \quad\cdots\quad b_n]^T$ where $x=a_1e_1+\cdots+a_ne_n$ and $y=b_1e_1+\cdots+b_ne_n$ are vectors with the $e_i$'s being the usual basis elements of $\mathbb{R}^n$.

Verify that this is an inner product.

• I am sure you want to verify that this is an inner product, and is E an nxn matrix? Nov 18 '11 at 4:24

I do not think this is an inner product unless E is an invertible matrix. Because if E is not invertible then there exist a non zero X in $\mathbb{R}^{nX1}$ such that $EX=0$, then ${\langle{X^T,X^T}\rangle}_E=0$ while $X^T\neq 0$.