1. ## inner product law

A is invertible real function,x is a vector
on what law
$(AA^tx,x)=(A^tx,A^tx)$

?

2. ## Re: inner product law

I presume you mean that A is a linear transformation- whether it is invertible is not really relevant. This is just the definition of $A^T$: If A is a linear transformation from vector space U to vector space V, then $A^T$ is defined as the linear transformation from V back to U such that $(Au, v)= (u, A^Tv)$ for any u in U and any v in V. The first inner product is in V and the second in U.

Starting from $(AA^Tx, x)$ with $u= A^Tx$ and $v= x$ that gives
$(AA^Tx, x)= (Au, v)= (u, A^Tv)= (A^Tx, A^Tx)$.

thanks