Mathematically, it is an expression generated by solving linear equations. The characteristics of determinant can be summarized as multiple alternating linear forms, which makes determinant a function describing "volume" in Euclidean space.

Basic properties of determinant

Properties of determinant of order n;

Property 1: Determinant is equal to its transposed determinant.

Property 2: exchange two rows (columns) of determinant, and the determinant changes sign.

Inference: If two rows in a determinant have the same corresponding elements (elements with the same column label), then the determinant is zero.

Property 3: The common factor k of a row in the determinant can be mentioned outside the determinant.

Inference: When two rows (columns) of elements in a determinant are proportional, the determinant is equal to zero.

Property 4: Determinant has branch (column) additivity.

Inference: If each element of a row (column) of a determinant is written as the sum of m numbers (m is an integer greater than 2), then this determinant can be written as the sum of m determinants.

Property 5: The elements in one row (column) of the determinant are multiplied by the same number and added to the corresponding elements in another row (column), and the determinant remains unchanged. [2]

Other attributes

If A is a reversible matrix, let A' be the transposed matrix of A (see * * * yoke). If the matrices are similar, their determinants are the same. Determinant is the product of all eigenvalues. This can be deduced from the similarity between matrix and Jordan canonical form.

Expansion of determinant

Cofactor (English translation)

Also known as "cofactor" and "cofactor". Refer to the principal cofactor. For an n-order determinant m, the determinant of n- 1 after removing the I-th row and the J-th column of m is called the sub-formula of m about the element mij. Write it down as Mij

The cofactor is cij = (- 1) (I+j) * mij.

Algebraic cofactor

In the determinant of order n, the determinant of order n- 1 after column j of row I where the (i, j) element aij is located is called the cofactor of the (i, j) element aij, and recorded as Mij: Note.

Aij=(- 1)i+j？ Mij

Aij is called the algebraic cofactor of (i, j)-ary aiji.

Expand rows and columns

An n-order determinant m can be written as the sum of the products of a row (or column) of elements and the corresponding algebraic cofactor, which is called a row (or column) expansion of the determinant.

This formula, also called Laplace formula, changes the calculation of determinant of order n into the calculation of determinant of order n- 1.

Determinant function

It can be seen from Laplace formula that the determinant of matrix A is a polynomial about its coefficients. Therefore, determinant function has good smoothness.

The determinant function of a single variable is set to the function of, as well. Its derivative of t is

The determinant function of a matrix is continuous. Therefore, the general linear group of order n is an open set, while the special linear group is a closed set.

Functions are also differentiable and even smooth (). Its development at a is as follows

That is to say, in the matrix space Mn () with regular norm, the adjoint matrix is the gradient of determinant function.

Especially when a is identity matrix,

The differentiability of invertible matrix shows that the general linear group GLn () is a Lie group.