Show that the matrix $AA^T+\alpha I$ is positive definite, where $\alpha >0$ and $A$ is an $m\times n$ real matrix.

Question

Show that the matrix $AA^T+\alpha I$ is positive definite, where $\alpha >0$ and $A$ is an $m\times n$ real matrix.

So I need to show that $x^T(AA^T+\alpha I)x>0$ for all vectors $x$. I'm really confused as to how to do this and how this is even true in general. Any solutions or hints are greatly appreciated.

Answer 1

Hint: $$ x^T(AA^T+\alpha I)x=x^TAA^Tx+\alpha x^Tx=|A^Tx|^2+\alpha|x|^2. $$

Answer 2

Hint:

1) Show that $AA^T$ is positive semidefinite using that $\langle Bx,x\rangle =\langle x,B^Tx\rangle$ and $\langle x,x\rangle =\|x\|^2_2$.

2) Show that $\alpha I$ is positive definite.

3) Show that the sum of positive definite and positive semidefinite matrices is positive definite.