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

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

v^{T}AA^{T}v = A^{T}v • A^{T}v ≥ 0.

I understand what *v*^{T}AA^{T}v 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*^{T}AA^{T}v is equivalent to *A*^{T}v • A^{T}v. I know that a^{T}b = a * •* b, but something else is going on, no? Appreciate any help!

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

a matrix M is said to be positive definite if for any vector v, v^{T}Mv ≥ 0

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

now v^{T}(A^{T}A)v = (Av)^{T}(Av) = (Av)•(Av).

(for any two matrices, it is a property of the transpose that (AB)^{T} = B^{T}A^{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^{T}A)v = (Av)•(Av) ≥ 0.

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

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

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.

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

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^{T}Mv ≥ 0, and v^{T}Mv = 0 if and only if v = 0.

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