Is matrix $AA^T + tI$ always regular?

Question

Is $AA^T + tI$, where $A \in \mathbb{R}^{m \times n}$ and $t > 0$, always regular for arbitrary $A$ and $t$?

Answer 1

If you consider the quadratic form $$\langle(AA^T + tI)x,x\rangle = \|A^Tx\|^2 + t \|x\|^2$$ you obverse that it the quadratic form is strictly positive for all $x \neq 0$. Therefore the Matrix $AA^T + tI$ has only positive eigenvalues which gives that the matrix is regular.