The Story about The Determinant

Hi folks,

I always had some difficulties understanding the concept of determinant. Whenever I read about it, and that was usually from Linear Algebra books, it was always described with a lot of "mathematical formality" and abstractness, and although I could follow and understand the definition and properties, the concept generally seemed to me like coming out of the blue.

"The best intuition" I have about determinant is that it measures how much does space get scaled by certain transformation. This intuition helped me a lot to understand some other concepts involving determinant, for example deriving the proof and explaining to myself why non-trivial solutions of Homogeneous System of Linear Equations occur when determinant of a matrix of the corresponding transformation equals 0. But, although I can prove this intuition holds for 2-dimension case, I find no way to generalize it for $n$ dimensions.

I feel that concept of determinant is really important to understand because it shows up in many branches of mathematics. I would really appreciate if someone could tell me a story about the determinant. I say story because it doesn't need to be highly mathematical and should aim to answer the following two questions:

Why did mathematicians introduced the concept of determinant? Why they needed it at the first place?
Based on that, how they derived the formula and generalized it for any number of dimensions?

Thanks in advance!

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2^{34}

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Remember to wrap math in `$` ... `$` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$

Comments

I don't have a story to tell, but I do have an anecdote. Suppose you have a matrix and a vector, such that the matrix pre-multiplied by the vector yields a vector of zeroes:

$\begin{pmatrix} a&b\\ c&d \end{pmatrix} \begin{pmatrix} x \\ y \end{pmatrix} = \begin{pmatrix} 0 \\ 0 \end{pmatrix}$

Suppose none of the elements $(a,b,c,d,x,y)$ are zero, for simplicity. What must the relationships between $a,b,c,d$ be?

$a x + b y = 0 \\ c x + d y = 0$

Rearranging gives:

$y = -\frac{a}{b} x$

Plugging into the second equation:

$c x - d \frac{a}{b} x = 0 \\ c - d \frac{a}{b} = 0 \\ b c - d a = 0 = a d - b c$

The quantity $(a d - b c)$ is well-known as the two-by-two matrix determinant. The determinant must be zero for the pre-multiplication of a vector by a matrix to yield a vector of zeros. It's almost as though the determinant is the matrix equivalent of a scalar magnitude.

Steven Chase - 2 years, 11 months ago

Proof which I derived for any $n \times n$ matrix and which I presented in my work.

"I will briefly prove here that non-trivial solutions of any homogeneous system of linear equations written in vector form $A\vec{x} = 0$ has non-trivial solutions when $\det(A) = 0.$ Let $i^{th}$ column entries of $A$ form vector $u_{i}$ so that: $A = \begin{bmatrix} a_{11} & a_{12} & \cdots & a_{1n} \\ a_{21} & a_{22} & \cdots & a_{2n} \\ \vdots & \vdots & \ddots & \vdots \\ a_{n1} & a_{n2} & \cdots & a_{nn} \end{bmatrix}= \begin{bmatrix} | & | & & | \\ u_{1} & u_{2} & \cdots & u_{n} \\ | & | & & | \\ \end{bmatrix}$

Then, by definition of matrix multiplication, we have:

$A\vec{x} = x_{1}u_{1} + x_{2}u_{2} + \cdots + x_{n}u_{n} = \theta,$

where $x_{i}$ is a scalar and, by conditions, at least one of them is non-zero. The above expression indicates that vectors $u_{1}, u_{2}, \cdots, u_{n}$ are linearly dependent. Ultimately, the determinant of a matrix formed by linearly dependent vectors is known to be zero. This comes from the fact that linear dependence implies that we can write some vector $u_{i}$ as a linear combination of other $n-1$ vectors ie. that $u_{i} \in \mathbb{R}^{k}$ where $k\leq n-1$ . Hence, the transformation defined by that matrix would squish the space to a lower dimension which is captured by zero determinant."

This comes from the intuition I mentioned: determinant is the quantitative measure of space transformation defined by a certain matrix.

You wrote about $\mathbb{R}^{2}$ case when calculations are pretty straightforward, but what about $\mathbb{R}^{3}$ , $\mathbb{R}^{4}$ , $\mathbb{R}^{n}$ space? I guess that the most elegant way would be to derive a proof using induction. But I'm wondering if that's actually possible?

Uros Stojkovic - 2 years, 11 months ago

Here's my idea of generalization of the formula by induction. Assume we have $n$ linear homogeneous equations and $n$ variables: $\begin{aligned}a_{11}x_{1} + a_{12}x_{2} + \cdots + a_{1n}x_{n} &= \sum_{k = 1}^{n}a_{1k}x_{k} = 0, \\ a_{21}x_{1} + a_{22}x_{2} + \cdots + a_{2n}x_{n} &= \sum_{k = 1}^{n}a_{2k}x_{k} = 0, \\ \vdots ~~~~~~~~~~~~~~~~&\vdots \\ a_{n1}x_{1} + a_{n2}x_{2} + \cdots + a_{nn}x_{n} &= \sum_{k = 1}^{n}a_{nk}x_{k} = 0.\end{aligned}$ This can be also represented in matrix form as $A\vec{x} = o$ . We can reduce this system to system of $n-1$ equations by substitution. Respectively: $x_{n}=-\frac{1}{a_{nn}} \sum_{k = 1}^{n-1}a_{nk}x_{k}.$ Now, plugging in back to the $j^{\text{th}}$ equation ( $1 \leq j \leq n-1$ ), we get: $\begin{aligned} a_{nn}\sum_{k = 1}^{n-1}a_{jk}x_{k} - a_{jn}\sum_{k=1}^{n-1}a_{nk}x_{k} = 0 \\ \sum_{k = 1}^{n-1}\left(a_{nn}a_{jk} - a_{jn}a_{nk} \right )x_{k} = 0\end{aligned}$

We are now left with one less equation and one less variable, but with new transformation matrix $B$ such that $b_{jk} = a_{nn}a_{jk} - a_{jn}a_{nk}$ . The idea is to prove general formula for evaluating determinant of $n \times n$ matrix by building upon the basis cases which are simple and intuitive.

Read up the history of the determinant; it is actually quite interesting. In any case, I think determinants arose as a consequence of solving a system of linear equations; if you have a system of $n$ linear equations in the variables $x_1, x_2, \dots, x_n$ and expressed them in terms of $y_1, y_2, \dots, y_n$ then inverting the system naturally gives you this quantity called a determinant. The area/volume/hypervolume story came much later, dating back to Euler and Tartaglia, but formalised by Cayley & Menger. If you want to see how Euler derived his story of volumes in terms of determinants, have a look at Dorrie's book 100 Great Problems in Elementary Mathematics (apologies in advance if I get the name wrong).

A Former Brilliant Member - 2 years, 10 months ago

Wonderful Youtube channel with intuitive videos. Here is their video on the determinant.

Andrei Li - 2 years, 9 months ago

Actually, I watched that video and all other from Linear Algebra series. I'm familiar with Grant Sanderson's content and I really enjoy it. However, except for the intuition I mentioned in my note above, the video doesn't offer an answer to my second question, namely, how is the formula derived for $n$ dimensions.

Uros Stojkovic - 2 years, 9 months ago

Oh, okay! No problem.

Math	Appears as
Remember to wrap math in `\(` ... `\)` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$