TITLE: Transpose matrix and inner product
QUESTION [7 upvotes]: In one of the proofs in class there was given the equality for the dot product:
$$\langle Ax, Ax\rangle = \langle x, A^tAx\rangle$$
I don't understand why this is correct. Is there a way to show this without explicitly looking at the multiplications and sums?
thanks.

REPLY [13 votes]: This is simpler than you might think.
By definition,
$$\langle x, y \rangle = x^{T}y\text{.}$$
Hence,
$$\langle Ax, Ax \rangle = (Ax)^{T}Ax$$
but by properties of matrix transposition, recall that 
$$(Ax)^{T}=x^{T}A^{T}$$
hence
$$\langle Ax, Ax \rangle = (Ax)^{T}Ax = x^{T}A^{T}Ax = (x)^{T}(A^TAx)=\langle x, A^TAx \rangle\text{.}$$