What does $\left|B\right|$ mean?

Question

Question: Suppose that $B$ is a matrix. What does (in general) $\left|B\right|$ mean?

Example: Let $B_{GS}$ be the iteration matrix for the Gauss-Seidel iterative method. I am given the following lemma:

Assume that $A\in\mathbb{R}^{n\times n}$ and assume $B_{GS}$ to be the Gauss-Seidel iteration matrix. Then $$\left|B_{GS}\right| \leq (I-L)^{-1}\left|U\right|.$$

Thanks in advance!

Answer 1

The inequality in the statement of the theorem eliminates the possibility that the notation $|A|$ refers to the determinant of $A$. It simply makes no sense to compare a real number to a matrix.

Given the context, it is likely that $|A|$ refers to the matrix $B = [b_{ij}]$ where $b_{ij} = |a_{ij}|$. The inequality should be interpreted in the componentwise sense, i.e. if $B$ and $C$ are matrices of the same size, then $B \leq C$ is the statement that $b_{ij} \leq c_{ij}$ for all values of $i$ and $j$.

I do not immediately recognize the validity of the theorem. The choice of $A=I$ leads to $L=I$ which renders the right hand side meaningless. It is clear that some significant restrictions are missing.

Answer 2

In general $|B|$ means $\det(B)$

Not sure what the following lemma means, but the notation is correct

So it's saying $\det(B_{GS})\leq (I-L)^{-1}\det(U)$, which makes sense since they are all just matrices.

It is possible for $|B|\lt 0$, this is not an absolute value symbol in this context