矩阵体积的一个基本性质

A property of the volume of matrix

矩阵体积是矩阵行列式绝对值的推广,也是向量长度的推广.在理解矩阵体积定义的基础上探讨两个矩阵乘积的体积与两个矩阵体积的乘积的大小关系,得出当两个矩阵乘积的秩与其中一个矩阵的秩t相等时,且另一个矩阵体积的平方大于或等于它的所有t阶子式的平方和,则两个矩阵乘积的体积小于或等于两个矩阵体积的乘积.

The volume of matrix is not only a promotion ot the absolute value ot determmant, Put also a promotion of the vector length. In this paper, on the size relations between the volume of two matrixes product and the product of two matrixes volume, based on the understanding of the definition of the matrix volume are studied, a result that the volume of two matrixes product is less than or equal to the product of two matrixes volume is obtained, when the rank of two matrixes product is equal to a matrix with rank t, and the square of another matrix volume is greater than or equal to the sum of the squares of its t-bands.