"A New Understanding of Matrices 5": Volume = Determinant

《新理解矩阵5》：体积=行列式 苏剑林 · 2013-12-25 · Mathematics algebra geometry matrix determinant

In the article "A New Understanding of Matrices 3": Bits and Pieces about Determinants, I first touched on the geometric meaning of the determinant — it represents the "volume" of an n-dimensional "parallelepiped." However, that article was written when I was just starting to learn about matrices, and some of the arguments there weren't rigorous, and even contained errors. Recently, while writing my end-of-term paper, I studied hypercomplex numbers, and the geometric meaning of determinants plays an important role in that work. So I revisited the geometric meaning of the determinant, fixed some of the earlier mistakes, and am sharing the corrected version here.

An $n$-th order matrix $A$ can be viewed as a collection of $n$ $n$-dimensional column vectors $\boldsymbol{x}_1,\boldsymbol{x}_2,...,\boldsymbol{x}_n$:

$$A=(\boldsymbol{x}_1,\boldsymbol{x}_2,\dots,\boldsymbol{x}_n)$$

From an algebraic point of view, this constitutes a matrix; from a geometric point of view, these $n$ vectors can be used to build a "parallel $n$-dimensional body." For instance: a parallelogram is a "parallel two-dimensional body," a parallelepiped is a "parallel three-dimensional body," and higher-order cases just follow by analogy — there's no need to actually visualize what a solid in high-dimensional space looks like. more

Let's consider the determinant $\det A$ of the matrix $A$. We know that $\det A$ has the following properties:

Properties of the determinant
1. The determinant is a function of $\boldsymbol{x}_1,\boldsymbol{x}_2,\dots,\boldsymbol{x}_n$, i.e., $\det A=f(\boldsymbol{x}_1,\boldsymbol{x}_2,\dots,\boldsymbol{x}_n)$;
2. (Linearity 1) If one column of the determinant is multiplied by a constant $\alpha$, then the value of the determinant is also multiplied by $\alpha$, i.e., $ f(\boldsymbol{x}_1,\dots,\alpha\boldsymbol{x}_i,\dots,\boldsymbol{x}_n)=\alpha f(\boldsymbol{x}_1,\dots,\boldsymbol{x}_i,\dots,\boldsymbol{x}_n)$；
3、（线性2）将行列式的某一列写成两列之和，那么行列式也相应地成为两个行列式之和，即$ f(\boldsymbol{x}_1,\dots,\boldsymbol{x}_i,\dots,\boldsymbol{x}_n)= f(\boldsymbol{x}_1,\dots,\boldsymbol{y}_i,\dots,\boldsymbol{x}_n)+f(\boldsymbol{x}_1,\dots,\boldsymbol{z}_i,\dots,\boldsymbol{x}_n)$，其中$\boldsymbol{x}_i=\boldsymbol{y}_i+\boldsymbol{z}_i$，性质二和三表明$f$是关于每个向量的线性函数；
4、（反对称）只要有两列相同，那么行列式值为0，即$f(\dots,\boldsymbol{x},\dots,\boldsymbol{x},\dots)=0$；
5、（归一）单位矩阵的行列式为1，即$f(I)=1$.

A remarkable fact is that the determinant is uniquely determined by the five properties above! That is, the five properties above uniquely determine a function $f$, and this function is precisely the determinant of the matrix.

From a geometric point of view, these $n$ vectors generate a parallel $n$-dimensional body in $n$-dimensional space. Let's consider the volume $V$ of this parallel $n$-dimensional body. We restrict our discussion to the first octant, in which case the volume satisfies the following properties (restricting to the first octant guarantees that all the vectors and scaling factors are positive).

Properties of volume
1. Volume is a function $V(\boldsymbol{x}_1,\boldsymbol{x}_2,\dots,\boldsymbol{x}_n)$ of these $n$ vectors;
2. If we multiply one vector by $\alpha$, that is, scale its length by a factor of $\alpha$, then the volume also increases by a factor of $\alpha$, i.e., $V(\boldsymbol{x}_1,\dots,\alpha\boldsymbol{x}_i,\dots,\boldsymbol{x}_n)=\alpha V(\boldsymbol{x}_1,\dots,\boldsymbol{x}_i,\dots,\boldsymbol{x}_n)$;
3. Volume is additive, i.e., $V(\boldsymbol{x}_1,\dots,\boldsymbol{x}_i,\dots,\boldsymbol{x}_n)= V(\boldsymbol{x}_1,\dots,\boldsymbol{y}_i,\dots,\boldsymbol{x}_n)+V(\boldsymbol{x}_1,\dots,\boldsymbol{z}_i,\dots,\boldsymbol{x}_n)$, where $\boldsymbol{x}_i=\boldsymbol{y}_i+\boldsymbol{z}_i$; this requires a bit of verification, but it is indeed true.
4. Whenever two of the vectors coincide, the volume is naturally 0, i.e., $V(\dots,\boldsymbol{x},\dots,\boldsymbol{x},\dots)=0$; for example, for a solid in three-dimensional space, if two of its edges coincide, then the solid has effectively been flattened into a plane, and the volume of a plane is naturally 0.
5. The parallel $n$-dimensional body formed by the identity matrix $I$ is a $n$-dimensional unit cube, whose volume is naturally 1, i.e., $V(I)=1$.

Comparing the properties of the determinant with those of volume, we see that they are exactly the same. So the volume of the parallel $n$-dimensional body in the first octant is precisely the determinant of the corresponding matrix! If we extend this to all octants, that's simply a generalization of the notion of volume (allowing it to be negative). Hence, we may as well define it this way: volume is the determinant.

In fact, introducing negative volume turns out to be quite significant — it is one of the foundations of what is now called "exterior differentiation." One of the classic uses of exterior differentiation is that it unifies results such as Gauss's divergence theorem and Stokes' theorem. It makes the theory and formalism of differentiation more complete and unified.