I am wondering whether matrix inequality induces matrix norm inequality for positive semidefinite matrix.
For example, considering two positive semidefinite matrices $A$ and $B$, when $A \succeq B$, does it imply $||A||\geq ||B||$($||\cdot||$ is any matrix norm)?
I think perhaps it is true for 2-norm. Since $A \succeq B$ implies $\lambda_k(A)> \lambda_k(B),\ \text{for}\ k=1,2...n$, where $\lambda_k$ is the k-largest eigenvalues. So, $\lambda_\max(A)> \lambda_\max(B)$.
Therefore, \begin{align*} \|A\|_{2} &= \sqrt{\lambda_{\max }\left(A^{*} A\right)} \\ &= \sqrt{\lambda_{\max }\left(A^{T} A\right)} \\ &= \sqrt{\lambda^2_{\max }\left(A\right)} \\ &\geq \sqrt{\lambda^2_{\max }\left(B\right)} \\ &= \sqrt{\lambda_{\max }\left(B^{T} B\right)} \\ &= \|B\|_{2}. \end{align*}
So, $A \succeq B \rightarrow \|A\|_{2}\geq ||B||_2$.
May I ask whether this reduction is correct? and does this hold for any other matrix norm?
Thanks in advance!
