Is there a way to measure the "magnitude" of a matrix?

Question

First in the case of unitary matrix $U^*=U^{-1}$. and $\det U=1$.

See more here: https://en.wikipedia.org/wiki/Unitary_matrix

However, notice $A=\begin{pmatrix} 0.1&0.1\\ 0.1&0.1\\ \end{pmatrix}$, $B=\begin{pmatrix} 10&10\\ 10&10\\ \end{pmatrix}$

Thus $\lim_{n\rightarrow\infty}A^n=\begin{pmatrix} 0&0\\ 0&0\\ \end{pmatrix}$, $\lim_{n\rightarrow\infty}B^n=\begin{pmatrix} \infty&\infty\\ \infty&\infty\\ \end{pmatrix}$, where $\lim_{n\rightarrow\infty}U^n$ was still a unitary matrix.

Obviously, $A$ converge to $0$ and $B$ blow up. In the sense that $|B|>|U|$ and $|U|>|A|$.

My question was that, is there any way to measure the magnitude of a matrix? (Notice $\det A=\det B=0$)

Answer 1

Yes. Take a look at the operator norm or in the case of matrices, the equivelant concept of matrix norm.