We are doing high school determinants and I have a doubt regarding the symbols used to represent modulus of a matrix.
I guessed it to be two pair of parallel lines viz. :
||A|| ,
where A is any square matrix.
But my teacher said it is wrong and instead suggested to write the modulus of a matrix in linguistic form i.e. mod det. A stating that ||A|| is used for something else.
After exploring I found that it is indeed used to represent the norm of a vector. But what's striking is that norm is similar to modulus (as I conclude after reading a bit of wikipedia).
Wikipedia page
So can anyone please help me understand the relation between the norm and modulus and also whether it can be used to represent modulus of a matrix in this form ||A||?
Thank You :)
- 11
-
4What do you mean by "modulus of a matrix" ? – SolubleFish Jun 14 '21 at 10:27
-
@SolubleFish From the context, it looks like it refers to the absolute value of the determinant – Ben Grossmann Jun 14 '21 at 22:53
-
@SolubleFish I meant determinant. I have edited it now. – Curiious Jun 17 '21 at 05:34
2 Answers
The notation that I would use is $|\det(\mathbf A)|$. A similar notation is typically used in the statement of the multivariate change of variables rule for integration, for instance.
If $\mathbf A$ is a matrix, then $\|\mathbf A\|$ typically denotes the norm of this matrix. For more information on matrix norms (in addition to the information you already found, see this Wikipedia page or my answer here. Note that the function $f(\mathbf A) = |\det(\mathbf A)|$ does not fit the definition of a matrix norm (since it lacks the homogeneity property), so I would say that the notation $\|\mathbf A\|$ for this function is inappropriate.
- 225,327
-
-
-
-
If $|\cdot|$ denotes a norm, then for any matrix $A$, $|A|$ is a number, as you said. – Ben Grossmann Jun 27 '21 at 18:05
There is a subtle typographical difference between $||A||$ and $\|A\|$ (absolute value of the determinant vs. norm).
There was no intent in this notation, it is purely coincidental. My bet is that to define a new norm operator, doubling the bars appeared naturally.
If you write a text where you need $||A||$ to appear, you should warn the reader that this is not a norm (or use Ben's notation). An ugly alternative is $|(|A|)|$.