Frobenius norm of two related matrices

Question

Given a matrix A, what is the relationship between the Frobenius norm of $A^TA$ and $A^TA - I$, where $I$ is the identity matrix. By relationship, I mean whether we can infer if one is bigger/smaller than one another, e.g., $\Vert A^TA\Vert_F \geq \Vert A^TA - I\Vert_F$.

Update: A is a real matrix

Any help would be appreciated.

Answer 1

Since $A$ is real then $X=A^TA$ is hermition and can be decomposed to $UDU^H$ where $U$ is unitary and $D$ diagonal with all the entries being real. Therefore we can write $$||X||_F=\sqrt{tr(XX^H)}=\sqrt{tr(DD^H)}=\sqrt{\sum_{i=1}^{n}d^2_{i}}$$therefore $$||X-I||_F=\sqrt{\sum_{i=1}^{n}(d_{i}-1)^2}$$so $$||X-I||_F\le||X||_F$$ if and only if$$\sum_{i=1}^{n}d_i\ge \dfrac{n}{2}$$

Answer 2

There is inequality between the two in general. Take $A=cI$ with $c>0$ small and $c$ large to get opposite inequalities.