In the article "New Understanding of Matrices 3: Bits of Determinants", I first discussed the geometric meaning of the determinant, noting that it represents the "volume" of an n-dimensional "parallelepiped." However, that article was written when I first began studying matrices, and some of the arguments were not rigorous or even contained errors. Recently, while writing my term paper, I researched topics related to hypercomplex numbers, where the geometric meaning of the determinant plays a crucial role. Consequently, I have re-examined the geometric interpretation of the determinant and corrected some errors. I am publishing this article to share these findings with everyone.
An n \times n matrix A can be viewed as a set of n column vectors \boldsymbol{x}_1, \boldsymbol{x}_2, \dots, \boldsymbol{x}_n in n-dimensional space: A = (\boldsymbol{x}_1, \boldsymbol{x}_2, \dots, \boldsymbol{x}_n) From an algebraic perspective, this constitutes a matrix; from a geometric perspective, these n vectors can establish an n-dimensional parallelepiped. For example, a parallelogram is a "2D parallelepiped," and a parallelepiped is a "3D parallelepiped." Higher-order cases can be understood by analogy without needing to actually visualize solids in high-dimensional space.
Let us consider the determinant of matrix A, denoted as \det A. We know that \det A possesses 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 a column of the determinant is multiplied by a constant \alpha, the value of the determinant is also multiplied by \alpha: 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. (Linearity 2) If a column of the determinant is written as the sum of two columns, the determinant is the sum of the two corresponding determinants: 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) where \boldsymbol{x}_i = \boldsymbol{y}_i + \boldsymbol{z}_i. Properties 2 and 3 indicate that f is a linear function with respect to each vector.
4. (Antisymmetry/Alternating) If any two columns are identical, the determinant is 0: f(\dots, \boldsymbol{x}, \dots, \boldsymbol{x}, \dots) = 0
5. (Normalization) The determinant of the identity matrix is 1: f(I) = 1
A remarkable fact is that the determinant is uniquely determined by the five properties above! That is, these five properties uniquely define a function f, which is the determinant of the matrix.
From a geometric perspective, these n vectors generate an n-dimensional parallelepiped in n-dimensional space. Let us consider the volume V of this n-dimensional parallelepiped. If we restrict our discussion to the first orthant (to ensure all vectors and factors are positive), the volume possesses the following properties:
Properties of Volume
1. Volume is a function of these n vectors: V(\boldsymbol{x}_1, \boldsymbol{x}_2, \dots, \boldsymbol{x}_n).
2. Multiplying a vector by \alpha (changing its length by a factor of \alpha) increases the volume by a factor of \alpha: 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: 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 point requires some verification, but it is indeed correct.
4. If any two vectors coincide, the volume is naturally 0: V(\dots, \boldsymbol{x}, \dots, \boldsymbol{x}, \dots) = 0 For instance, if a solid in 3D space has two edges that coincide, the solid has been compressed into a plane, and the volume of a plane is naturally 0.
5. The n-dimensional parallelepiped formed by the identity matrix I is an n-dimensional unit cube, and its volume is naturally 1: V(I) = 1
Comparing the properties of the determinant with those of volume, we find they are identical. Therefore, the volume of an n-dimensional parallelepiped in the first orthant is the determinant of the corresponding matrix! If we extend this to all orthants, it is simply a generalization of the concept of volume (allowing for negative values). Thus, we might as well define it this way: Volume is the determinant.
In fact, the introduction of negative volume is of great significance; it is one of the foundations of modern “exterior calculus.” A typical use of exterior calculus is its ability to unify the Gauss divergence theorem, the Stokes theorem, and others. It makes the theory and form of calculus more complete and unified.
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