0
$\begingroup$

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

$\endgroup$
2
  • 1
    $\begingroup$ Yes. ${}{}{}{}{}$ $\endgroup$ Jan 15, 2020 at 16:35
  • $\begingroup$ You've asked a sensible mathematical question, but you provided no context. Did this come up in the solution of a larger problem, or was it a standalone assigned exercise? Providing context, such as showing you approached the problem from some basic idea (e.g. applying the definition), will not only help Readers respond in a useful way now, it will help future Readers benefit from your post. $\endgroup$
    – hardmath
    Jan 3, 2022 at 15:23

1 Answer 1

3
$\begingroup$

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.

$\endgroup$

Not the answer you're looking for? Browse other questions tagged .