In mathematics, determinant is a function of matrix A whose domain is det, and its value is scalar, which is denoted as det(A) or | A |. Whether in linear algebra, polynomial theory or calculus (such as substitution integral method), determinant, as a basic mathematical tool, has important applications.
Determinant can be regarded as a generalization of the concept of directed area or volume in general Euclidean space. In other words, in N-dimensional Euclidean space, determinant describes the influence of a linear transformation on "volume".
mathematic definition
N-order determinant
set up
Is made up of n arranged in an n-order square matrix? The number aiji (I, j = 1, 2, ..., n), whose value is n! Sum of terms
Where k 1, k2, ..., kn are exchanged by the sequence 1, 2, ..., n represents k times, and σ represents k 1, k2, ..., kn. The form is as follows
The sum of the items of, where a 13a2 1a34a42 corresponds to k=3, that is, the symbol before the item should be
(- 1)3.
If the n-order square matrix A=(aij), then the determinant D corresponding to A is recorded as
D=|A|=detA=det(aij)
If the determinant d of matrix A is equal to 0, it is called singular matrix, otherwise it is called nonsingular matrix.
Tag set: any k elements I 1, I 2, ..., n are satisfied in the sequence i 1, i2, ..., ik.
1≤I 1 & lt； i2 & lt...& ltik≤n( 1)
I 1, i2, ..., ik has k {1, 2, ..., n} and {1, 2, ..., n} represented as C. Subcolumns. Therefore, C(n, k) is a label set with elements, and the elements of C(n, k) are expressed as σ, τ, ..., σ∈C(n, k).
σ={i 1，i2，...，ik}
It's {1, 2, ..., n}. If τ={j 1, j2, ..., jk}∈C(n, k), then σ = τ means i65438.
nature
① A row (or a column) in determinant A is multiplied by the same number k, and the result is equal to kA.
② determinant a is equal to its transposed determinant at (the I-th row of at is the I-th column of a).
③ If there is a row (or a column) in the determinant of order n |αij|; The determinant |αij| is the sum of two determinants, where the first row (or column) is B 1, B2, ..., bn; The other is с 1, с 2, …, с n; The elements in other rows (or columns) are exactly the same as those in |αij|.
④ Two rows (or columns) in determinant A are interchanged, and the result is equal to -a..⑤ Multiply each element in one row (or column) of determinant A by a number, and then add it to each corresponding element in another row (or column), and the result is still A. ..
What is a determinant?
Determinant is a function in mathematics, which maps a matrix A into a scalar, marked as det(A) or | A |. Determinant can be regarded as a generalization of the concept of directed area or volume in general Euclidean space. In other words, in N-dimensional space, determinant describes the influence of a linear transformation on "volume". Whether in linear algebra, polynomial theory or calculus (such as substitution integral method), determinant, as a basic mathematical tool, has important applications.
The concept of determinant first appeared in the process of solving linear equations. /kloc-In the late 7th century, determinant has been used in the works of Guan Xiaohe and Leibniz to determine the number and form of solutions of linear equations. Since18th century, determinant has been studied as an independent mathematical concept. /kloc-After the 9th century, the determinant theory has been further developed and perfected. With the introduction of the concept of matrix, more properties of determinant have been discovered. Determinant has gradually shown important significance and function in many fields, and the definitions of linear endomorphism and determinant of vector groups have appeared.
The characteristics of determinant can be summarized as multilinear form, which makes determinant a function describing "volume" in Euclidean space.
Vertical line symbol of determinant
The determinant of matrix A is sometimes called |A|. Absolute value and norm | matrix norm also use this notation, which may be confused with determinant notation. However, matrix norms are usually represented by double vertical lines (such as:), and subscripts can be used. Furthermore, the absolute value of the matrix is undefined. Therefore, determinants often use vertical notation (for example, Cramer's rule and sub-table). For example, the matrix:
,
The determinant det(A) is also written as | A |, or explicitly written as:
That is, the square brackets of the matrix are replaced by slender vertical lines.
The history of determinant
The concept of determinant was originally developed with the solution of the equation. Determinant can be traced back to17th century at the earliest, and its initial prototype was developed by Japanese mathematician Guan Xiaohe and German mathematician gottfried? Leibniz got it independently, and the time was about the same.
Early research on determinant
Guan Xiaohe used the concept of determinant for the first time in solving problems. 1545, cardan gave a method to understand two linear equations in his book The Tree. He called this method "mother method". This method is very similar to the later Cramer's rule, but Cardan did not give the concept of determinant.
1683, Japanese mathematician Guan Xiaohe introduced the concept of determinant for the first time in his book Method of Solving Problems. The determinant of, even in the book, is used to solve higher-order equations.
1693, the German mathematician Leibniz began to use the index set of the system to represent the coefficients of three ternary linear equations. He got a determinant by eliminating two unknowns from three equations. This determinant is not equal to zero, which means that a set of solutions satisfies three equations at the same time. [5] Because there was no concept of matrix at that time, Leibniz used number pairs to represent the position of elements in the determinant: representing the I-th row and the J-th column. Leibniz's research achievements on determinant have included the expansion of determinant and Clem's law, but these achievements were not known at that time.
Arbitrary determinant
1730, Scottish mathematician Colin? In his Algebra, maclaurin began to expound the theory of determinant, recorded the methods of solving binary, ternary and quartic linear equations with determinant, and gave the correct form of the general solution of quartic linear equations, although this book was published two years after maclaurin's death (1748).
1750, Gabriel? In his Introduction to Algebraic Curve Analysis, Kramer first gave a rule to solve the system of linear equations with n variables, which was used to determine the coefficients of the general quadratic curve passing through five points, but it was not proved. [8] The calculation of determinant is very complicated, because it is defined in the parity of permutation.
Since then, the research on determinant has gradually increased. France Etienne 1764? The research on the calculation method of determinant in Bezu's paper simplifies Clem's law and gives a method to distinguish linear equations with the resultant [10]. Alexander, who is also French? Theophile? Vandermonde, was the first to separate determinant from equation theory in his paper 177 1, and expounded determinant separately. This is the beginning for mathematicians to study determinant itself.
1772, Pierre Simon? In the article "Discussion on Integral and World System", Laplace extended the method of expanding the determinant in Vandermonde,'s works into the sum of several smaller determinants, and developed the concept of sub-formula. A year later, Joseph? Lagrange discovered the relationship between determinant and space volume. He found that the volume of the tetrahedron formed by the origin and three points in space is one sixth of the determinant formed by their coordinates.
Determinant is called "deterministic" in most European languages (in some languages, e or o is added to the suffix, or it becomes s), which was first developed by Karl? Friedrich? Gauss introduced it in his arithmetic research. The root of this name means "determination", because in the use of Gaussian, determinant can determine the properties of conic. In the same book, Gauss also described a method to solve multivariate linear equations by adding and subtracting coefficients, which is now gauss elimination.
Modern concept of determinant
After entering the19th century, the determinant theory has been further developed and improved. Augustine? Louis? Cauchy first used the word "certainty" to express the determinant that appeared in 18 12 years in 18 century. In the past, Gauss only limited this word to the coefficient determinant corresponding to the quadratic curve. Cauchy was also the first mathematician to arrange determinants into a square matrix and express its elements with double subscripts (the vertical symbol is Arthur? Cauchy, first used by Gloria in 184 1, also proved the nature of determinant (actually matrix multiplication), Jacques? Philip? Mary? It appeared in Binet.Alfred's book, but it has not been proved.
1In the 1950s, Gloria and James? Joseph? Sylvester introduced the concept of matrix into mathematical research [12]. The close relationship between determinant and matrix makes the matrix theory flourish, but it also brings many new results about determinant, such as Hadamard inequality, orthogonal determinant, symmetric determinant and so on.
At the same time, determinant has also been applied in various fields. When Gauss studied quadratic curve and quadratic form, he regarded determinant as the standard form of quadratic curve and quadratic form. After that, Karl Weierstrass and Sylvester perfected the theory of quadratic form and studied the failure of analysis (PNG conversion failure; Please check whether latex, dvips, gs and convert): \lambda matrix determinant and elementary factor are installed correctly. Determinants have been used to integrate various functions since the 1930s. Between 1832 and 1833, Carl? Jacoby found some special results, 1839, Eun? Charles. Catalonia discovered the so-called Jacobian determinant. 184 1 year, Jacobi published a paper on function determinant, and discussed the relationship between linear correlation of function and Jacobian determinant.
The concept of determinant in modern sense was first introduced to China at the end of 19. 1899, Hua and John Flair, a British missionary, jointly translated fourteen volumes of Calculation Method, in which determinant was translated into "fixed number" for the first time. 1909, Gu Cheng called it "fixing the column" in his works. 1935 In August, chinese mathematical society reviewed the translation of various terms, and in September, the Ministry of Education published mathematical terms, formally defining translation as "determinant". Since then, "determinant" has been used as a translation name.
Intuitive definition of determinant
The determinant of the n-order block matrix A can be intuitively defined as follows:
Where Sn is the set {1, 2, ..., n}, that is, from the set {1, 2, ..., n} to itself;
Represents the sum of all elements of s, that is, for each σ∈S, it appears once in the addition formula; For each number pai, jr (i, j), ai and j that satisfies 1 ≤ i and j ≤ n, they are the elements in the i-th row and the j-th column of matrix A.
σ represents the parity of the permutation σ∈Sn, specifically, it satisfies1≤ i.
If * * * has an even number of σ inverses, then sgn(σ) = 1, and if * * * has an odd number, then sgn(σ) =? 1。
For example, for 3 yuan permutation σ = (2,3, 1) (i.e. σ( 1)=2, σ(2)=3, σ(3)= 1, because 1 is after 2, 1. But for ternary permutation σ = (3,2, 1) (that is, σ=3, σ=2, σ= 1), it can be calculated that * * has three reverse orders (odd numbers), so sgn(σ) =? 1, so that the symbols of a 1, 3a2, 2a3, 1 in the third-order determinant are all negative signs.
Note that for any positive integer n, S_n*** has n elements, so * * * in the above formula has n summation terms, that is, this is a finite summation term.
For simple 2-order and 3-order matrices, the expression of determinant is relatively simple, which is exactly the sum of the product of each major diagonal (upper left to lower right) minus the product of each minor diagonal (upper right to lower left) (see red line and blue line in figure 1).
σ represents the parity of the permutation σ∈Sn, specifically, it satisfies1≤ i.
If * * * has an even number of σ inverses, then sgn(σ) = 1, and if * * * has an odd number, then sgn(σ) =? 1。
For example, for 3 yuan permutation σ = (2,3, 1) (i.e. σ( 1)=2, σ(2)=3, σ(3)= 1, because 1 is after 2, 1. But for ternary permutation σ = (3,2, 1) (that is, σ=3, σ=2, σ= 1), it can be calculated that * * has three reverse orders (odd numbers), so sgn(σ) =? 1, so that the symbols of a 1, 3a2, 2a3, 1 in the third-order determinant are all negative signs.
Note that for any positive integer n, S_n*** has n elements, so * * * in the above formula has n summation terms, that is, it is a finite summation term.
For simple 2-order and 3-order matrices, the expression of determinant is relatively simple, which is exactly the sum of the product of each major diagonal (upper left to lower right) minus the product of each minor diagonal (upper right to lower left) (see red line and blue line in figure 1).
Determinant of second-order matrix;
Determinant of a third-order matrix;
However, for the square matrix A of order n≥4, there are only n such major diagonals and minor diagonals, because the total number of major diagonals and minor diagonals of A is 2n.
So in addition to this diagonal product, there are more items added to the determinant. For example, in the fourth-order determinant, the terms a 1, 2a2, 3a3,1a4,4 are not the product of any diagonal elements. However, as in the case of 2-order and 3-order determinants, every item in the n-order determinant is still obtained by multiplying n elements of the matrix, and only one element is selected in each row and column, and the whole determinant traverses all such selection methods just once.
In addition, each row or column of an n×n matrix can also be regarded as an n-ary vector, and the determinant of the matrix is also called the determinant of the vector group composed of the n-ary vectors.