行列式的几何意义

01 Jun 2020

Given a square matrix:

\[\mathbf{A}=\left[\begin{array}{ccc} & \mathbf{a}_{(1)}^{T} & \\ & \mathbf{a}_{(2)}^{T} & \\ & \vdots & \\ & \mathbf{a}_{(n)}^{T} & \end{array}\right]\]

consider the set of points S ⊂ Rⁿ formed by taking all possible linear combinations of the row vectors, where the coefficients of the linear combination are all between 0 and 1; that is, the set S is the restriction of span({a₁, …, a_n}) to only those linear combinations whose coefficients satisfy 0 ≤ α_i ≤ 1, i = 1, . . . , n. Formally

\[\mathcal{S}=\left\{v \in \mathbb{R}^{n}: v=\sum_{i=1}^{n} \alpha_{i} \mathbf{a}_{i}, \text { where } 0 \leq \alpha_{i} \leq 1, i=1, \ldots, n\right\}\]

For example

\[\mathbf{A}=\left[\begin{array}{ll} 1 & 3 \\ 3 & 2 \end{array}\right]\]

the row vectors of A is

\[\mathbf{a}_{1}=\left[\begin{array}{l} 1 \\ 3 \end{array}\right] \quad \mathbf{a}_{2}=\left[\begin{array}{l} 3 \\ 2 \end{array}\right]\]

The set S corresponding to these rows is shown in figure below, In our example, the value of the determinant is det(A) = −7, so the area of the parallelogram is 7.

也就是说，矩阵行列式的绝对值等于此面积。

多维的情况是一样的。三维的情况也容易想象(包围的体积)。