Thread: help me understand a line in an "ATA is positive, semi-definite" proof

I am looking at a proof for why A^TA is positive semi-definite when A is nxn and it has this line.

v^TAA^Tv = A^Tv • A^Tv ≥ 0.

I understand what v^TAA^Tv means and the purpose of proving that it's nonnegative, etc... My problem is that I am a linear algebra novice and do not necessarily understand how the first part v^TAA^Tv is equivalent to A^Tv • A^Tv. I know that a^Tb = a • b, but something else is going on, no? Appreciate any help!

a matrix M is said to be positive definite if for any vector v, v^TMv ≥ 0

note that the inner (dot) product of two vectors u,v can be written as the matrix product u^Tv.

now v^T(A^TA)v = (Av)^T(Av) = (Av)•(Av).

(for any two matrices, it is a property of the transpose that (AB)^T = B^TA^T).

but for any vector u, u•u ≥ 0 (this is a property of inner products, called positive-definiteness as well),

hence v^T(A^TA)v = (Av)•(Av) ≥ 0.

(by the way, the proof you show is one that shows AA^T is positive definite).

Oh great - I figured I was just making a dumb mistake in parsing vT(ATA)v.

Now you've confused me about something else, though. I thought it was semi definite if greater than or equal to 0 and definite if only greater than 0? This seems like the consensus just from googling and Wikipedia confirms.

well, the thing is, v might be the 0-vector. if v is NOT the 0-vector, then the same exact proof shows we are strictly greater than 0. the precise definition should be:

M is positive definite if v^TMv ≥ 0, and v^TMv = 0 if and only if v = 0.

ok thanks a lot!