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

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

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

vTAATv = ATv • ATv ≥ 0.

I understand what vTAATv 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 vTAATv is equivalent to ATv • ATv. I know that aTb = a b, but something else is going on, no? Appreciate any help!

2. ## 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, vTMv ≥ 0

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

now vT(ATA)v = (Av)T(Av) = (Av)•(Av).

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

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

hence vT(ATA)v = (Av)•(Av) ≥ 0.

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

3. ## 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.

4. ## 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 vTMv ≥ 0, and vTMv = 0 if and only if v = 0.

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

ok thanks a lot!