# Why are |vertical lines| used to mark matrix determinants?

This notation is sometimes used to denote the determinant: $$\begin{vmatrix}a & b \\ c & d\end{vmatrix} = ad-bc$$ Why? Where did this notation come from? Was there any relationship between this notation and the absolute value $$|x|$$ or the norm $$\lVert\mathbf{x}\rVert$$?

• – lhf
May 10, 2019 at 16:08

In 1841, Cayley published the first English contribution to the theory of determinants. In this paper he used two vertical lines on either side of the array to denote the determinant, a notation which has now become standard.

Update: Thanks to @HansLundmark for the reference to Cajori's, A History of Mathematical Notations, section 462. Modern Notations, where the text states:

"-A notation which has rightly enjoyed great popularity because of its objective presentation of the elements composing a determinant, in convenient arrangement for study, was given in 1841 by Cayley..."...

"The first occurrence of Cayley's vertical-line notation for determinants and double vertical-line notation for matrices in Grelle's Journall is in his "Memoirs sur les hyperdeterminants" ; in Liouville's Journal, there appeared in 1845 articles by Cayley in which [ ] and { } are used in place of the vertical Iines.! The notation { } was adopted by O. Terquem" in 1848, and by F. Joaehimsthal' in '1849, who prefixes "det," thus: "det, { }." E. Catalan! wrote "det. (A, B, G .... )," where A, B, G, .•.. , are the terms along the principal diagonal. The only objection to Cayley's notation is its lack of compactness. For that reason, compressed forms are used frequently when objective presentation of the elements is not essential...."

• That was fast, considering no amount of my google queries could answer this question. But still, ...why? May 10, 2019 at 16:09
• @Infiaria: In Mathematics, many writers choose a shorthand and sometimes, it just becomes the standard, as in this case - it was just his personal preference. Sometimes this is good and sometimes bad as different authors change the notation or invent their own (no issue with that if it is properly defined) and we have too many ways to define something, which is hard for new users. In this case however, everyone has accepted the shorthand.
– Moo
May 10, 2019 at 16:20
• I suppose so, but considering writing matrices is arduous anyway (especially in latex), I'm not sure if writing $\det$ instead would be that much more trouble. May 10, 2019 at 16:22
• @Infiaria: Imagine how hard it was when they only had a printing press. If you ever look at old math books (many online), it is amazing that they were able to do what they did with the limited tools at their disposal - many of the books are very difficult to read by today's standards. Compare a calculus or trigonometry text from the 1800's to one today, for example. We are so spoiled with all of the tools at our disposal.
– Moo
May 10, 2019 at 16:37
• Confirmed by Cajori's A History of Mathematical Notations, section 462. See also here: www-history.mcs.st-and.ac.uk/HistTopics/… May 10, 2019 at 17:07

There is a relationship between the vertical line notation for determinant and the notation $$|x|$$ for absolute value and $$||\mathbf{x}||$$ for norm, however I do not know whether this was an intentional decision historically. The absolute value, norm, and determinant, all have at least two things in common.

1. They are functions mapping a given quantity (a real number, a vector, or a matrix) to a real number.

2. They measure the size of something. The absolute value and norm give the distance from the origin to the real number or vector. And the determinant is the factor by which the volume of the unit cube increases under the linear transformation represented by the matrix.

One catch with the analogy is that unlike absolute value and norm, determinants can be negative. In this case however, they are still measuring the factor of volume change. A negative sign simply indicates a change in orientation.

• I was wondering if the reasoning for the lines because it was similar to the ideas of "magnitude" that $|x|$ and $\lVert\mathbf{x}\rVert$ has but I considered otherwise because (a) vectors are often considered matrices of size $n\times 1$ (or opposite), of which the vector norm can be calculated and the determinant not (as they are not square), and (b) it seems there is already a "matrix norm" but the topics mentioned in there are out of my scope of understanding. May 10, 2019 at 17:40
• Both are good points. The matrix norm is like a length, while the determinant is like a volume. For example, the Frobenius norm equals the square root of the sum of the squares of the eigenvalues, which suggests a Euclidean length, while the determinant is the product of the eigenvalues, which suggests a volume. Since volume and length are related geometric notions, I think using vertical line notation for either matrix norms or determinants is reasonable. This is just a possible justification. I don't know what, if any, reasons there were historically for the notation. May 10, 2019 at 20:09
• @CoolerParadox That is something I never thought of Thanks. Jul 19, 2023 at 12:23