What is the meaning of determinant?
The determinant is equal to the volume of the parallel 2n-hedron formed by the vectors corresponding to its columns. This is because the determinant is an alternating multilinear form, and the volume in the Euclidean space we usually understand is also such a function (the volume of the unit cube is 1. If you expand c times along a side, the volume will be expanded by c times. After exchanging two sides, the volume will be reversed. This is a supplementary definition. We think that the volume is a directed volume, and its value indicates the size of the volume, and the positive and negative signs indicate The arrangement order of each side or the chirality of the coordinate axis), and the function that satisfies the normalization, multi-linearity, and anti-symmetry is unique, so the intuitive understanding of the determinant is the directed volume in the Euclidean space.
It is also easy to understand by using the isomorphism of matrix and linear transformation to explain. The determinant is the measure of the degree of stretching of the space by the linear transformation corresponding to the matrix, or the volume ratio of the object before and after transformation. In particular, if the matrix is not of full rank, it means that an nnn-dimensional space is transformed and flattened, becoming one of the n−1n-1n-1 dimensional hyperplanes or even lower-dimensional hyperlinear lines, Therefore, the volume element in the original space is 0 after transformation, and the determinant is also 0 at this time. This is the reason why multivariate function integration needs to be multiplied by a Jacobi determinant after variable substitution, which represents the volume ratio of microelements before and after transformation.
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determinant of matrix
