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Determinants

I just feel like ranting about determinant notation. I always get in this mood when preparing a lecture on determinants and I look through various books for ideas on better presentation and the somewhat standard notation makes my skin crawl. Many people think it is a good idea to use

$\left\lvert \begin{matrix} a & b \\ c & d \end{matrix} \right\rvert$

instead of the sane, and hardly any more verbose

$\det \left[ \begin{matrix} a & b \\ c & d \end{matrix} \right]$ or $\det \left( \left[ \begin{matrix} a & b \\ c & d \end{matrix} \right] \right)$ .

Now what’s the problem with the first one.

1) Unless you look carefully you might mistake the vertical lines for brackets and simply see a matrix, not its determinant.

2) vertical lines look like something positive while the determinant is negative.

3) What about 1 by 1 matrces. $|a|$ is a determinant of $[a]$ or is it the absolute value of $a$ .

4) What if you want the absolute value of the determinant (something commonly done). Then if you’d write

$\left\lvert\left\lvert \begin{matrix} a & b \\ c & d \end{matrix} \right\rvert\right\rvert$

that looks more like the operator norm of the matrix rather than absolute value of its determinant. So in this case, even those calculus or linear algebra books that use the vertical lines will write:

$\left\lvert \det \left( \left[ \begin{matrix} a & b \\ c & d \end{matrix} \right] \right) \right\rvert$

So now the student might be confused because they don’t expect to see “det” used for determinant (consistency in notation is out the window).

So … if you are teaching linear algebra or writing a book on linear algebra, do the right thing: Don’t use vertical lines.

Syndicated 2012-05-09 20:51:55 from The Spectre of Math

9 May 2012 vicious » (Master)