Determinant of $\lambda I + A^TA$

Question

What properties $\lambda I + A^TA$ have? I know that $A^T A$ is positive semi-definite, and symmetric. I want to show that the determinant of $\lambda I + A^TA$ decreases as $\lambda$ increases!

Answer 1

After an orthogonal change of basis $A^TA$ becomes a diagonal matrix $D$ with nonnegative entries $d_1,\dots,d_n$. The question is then whether $$ f(\lambda) = \det(\lambda I+D) = \prod_{i=0}^n(\lambda+d_i) $$ increases or decreases as $\lambda$ increases. For $\lambda\geq0$, we see that each term in the product is positive and increases with $\lambda$ so $f(\lambda)$ increases as $\lambda$ increases. (That is, $f'(\lambda)>0$ for $\lambda>0$.)

But it is possible that $f$ is decreasing for sufficiently negative arguments. For example, if $n$ is even and $D=0$, you can see that $f(\lambda)=\lambda^n$ is a decreasing function in the area $\lambda\leq0$.