My text book is called "Linear Algebra and its applications" by David C. Lay.

I am just wondering why the textbook uses the absolute value symbol when it wants us to compute determinants. For example, for some matrix A, the determinant is represented as |A| (Chapter 3.1 Determinants).

And yet 2 sections later (3.3 Volume), we actually use absolute values around determinants, which is represented like this: | det(A) |. So if we wanted to compute the determinant of Matrix A as well as have the absolute value of it, it would look like ||A||?

I just don't see why the text book would choose to denote determinants using the same symbols as absolute values. Why would they not choose a different representation of determinants?

  • $\begingroup$ The two bars used to be the most common notation. With the fading of importance of the determinant, $\det$ has become more common. $\endgroup$ – André Nicolas Oct 25 '12 at 18:45
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    $\begingroup$ @AndréNicolas What do you mean by the "the fading importance of the determinant"? Has the use of determinants become less prominent nowadays? $\endgroup$ – EuYu Oct 25 '12 at 18:48
  • $\begingroup$ It is historical, but rarely confusing in context. $\endgroup$ – copper.hat Oct 25 '12 at 18:54
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    $\begingroup$ @EuYu: Much less. It used to be that Cramer's Rule was the standard algorithm students were given to solve a system of linear equations! Even though calculating determinants is computationally unpleasant. For quite a few years now, some variant of Gaussian elimination has been used instead in introductory courses. There was in the old days a greater taste for "formulas." $\endgroup$ – André Nicolas Oct 25 '12 at 18:55
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    $\begingroup$ @AndréNicolas Regardless of which method is used in introductory courses to solve linear systems, determinants are still really important! I would not want to do exterior algebras and differential forms without determinants. Or find characteristic polynomials and eigenvalues, for that matter. $\endgroup$ – Aleksandar Bahat Oct 25 '12 at 19:27

The notation $|A|$ is used to denote the determinant of the matrix $A$. This is especially useful if you are currently calculating the determinant, for example $$\begin{vmatrix}3 & 4 \\ 2 & 1\end{vmatrix} = \det\begin{pmatrix}3 & 4 \\ 2 & 1\end{pmatrix}$$ The notation $\|A\|$ is typically the norm of the matrix for whatever norm you are using. On the otherhand, $\det(A)$ is another notation for the determinant of the matrix $A$. If you want the absolute value of the determinant, I would use the same notation as your textbook and use $|\det(A)|$. I would not suggest using $\|A\|$ do denote the absolute value of the determinant since there's a confusion with the norm.

To be fair to the textbook, there really isn't a chance of confusion. There is no such thing as the "absolute value" of a matrix. $|A|$ always means determinant.

  • $\begingroup$ Your last part of your answer states that there is no such thing as an absolute value of a matrix. So let's say we have matrix A, which is a system of equations representing lines in 2 space. We can definitely take the absolute value of line equations, so is there no operation that takes the absolute values of all the line equations in the matrix? $\endgroup$ – krikara Oct 25 '12 at 18:52
  • $\begingroup$ Well, it kind of depends on what you mean by "absolute value". The term absolute value is usually reserved for real numbers, in which case it means the absolute magnitude of the number. What exactly is the "absolute value" of a line? Concepts similar to the absolute value is developed in matrix analysis and the notion of a norm is used to give a representation to the "size" or "magnitude" of a matrix. But even then, different norms will have different interpretations on exactly what "size" means. $\endgroup$ – EuYu Oct 25 '12 at 18:56

It's simple convenience. The determinant of a matrix is at least a bit like an absolute value, since matrices that expand volumes have determinants greater than $1$ and vice versa. But it's not properly an absolute value, norm for those who know the word, since lots of nonzero matrices have zero determinant.

To further complicate things, in analysis courses you'll begin to encounter proper absolute values on matrices, the simplest of which is just the matrix's length as a vector in $\mathbb{R}^{n^2}$. Another is the operator norm, which is most often denoted exactly by $||A||$. For that reason and for clarity more generally you should never use, nor see, $||A||$ to mean $|\det A|$.


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