행렬식

개요

• 선형대수학과 행렬이론의 주요 개념
• 유클리드 공간에서의 부피 개념
  • 유클리드 평면의 2차원 벡터 두 개가 만드는 평행사변형의 넓이
  • 유클리드 공간의 3차원 벡터 세 개가 만드는 평행육면체의 부피
• 교대 다중선형형식의 예

정의

• n x n 행렬 \(A=(a_{ij})_{1\le i,j \le n}\)에 대하여, 다음과 같이 행렬식을 정의

\[\det(A) = \sum_{\sigma \in S_n} \operatorname{sgn}(\sigma) \prod_{i=1}^n a_{i \sigma(i)}\] 여기서 \(S_n\)은 대칭군 (symmetric group)

• 행렬 \(A=(a_{ij})\)의 행렬식을 \(|a_{i,j}|_{1\le i,j \le n}\) 형태로 표현하기도 함

예

• \(n=1\) 일 때,

\[ \begin{vmatrix} a_{1,1} \end {vmatrix} =a_{1,1} \]

• \(n=2\)일 때,

\[ \begin{vmatrix} a_{1,1} & a_{1,2} \\ a_{2,1} & a_{2,2} \end {vmatrix} =a_{1,1} a_{2,2}-a_{1,2} a_{2,1} \]

• n=3일 때,

\[ \begin{vmatrix} a_{1,1} & a_{1,2} & a_{1,3} \\ a_{2,1} & a_{2,2} & a_{2,3} \\ a_{3,1} & a_{3,2} & a_{3,3} \end {vmatrix} =a_{1,1} a_{2,2} a_{3,3}-a_{1,1} a_{2,3} a_{3,2},-a_{1,2} a_{2,1} a_{3,3}+a_{1,2} a_{2,3} a_{3,1}+a_{1,3} a_{2,1} a_{3,2}-a_{1,3} a_{2,2} a_{3,1} \]

• n=4일 때,

\[ \begin{vmatrix} a_{1,1} & a_{1,2} & a_{1,3} & a_{1,4} \\ a_{2,1} & a_{2,2} & a_{2,3} & a_{2,4} \\ a_{3,1} & a_{3,2} & a_{3,3} & a_{3,4} \\ a_{4,1} & a_{4,2} & a_{4,3} & a_{4,4} \end {vmatrix} =a_{1,4} a_{2,3} a_{3,2} a_{4,1}-a_{1,3} a_{2,4} a_{3,2} a_{4,1}-a_{1,4} a_{2,2} a_{3,3} a_{4,1}+a_{1,2} a_{2,4} a_{3,3} a_{4,1}+a_{1,3} a_{2,2} a_{3,4} a_{4,1}-a_{1,2} a_{2,3} a_{3,4} a_{4,1}-a_{1,4} a_{2,3} a_{3,1} a_{4,2}+a_{1,3} a_{2,4} a_{3,1} a_{4,2}+a_{1,4} a_{2,1} a_{3,3} a_{4,2}-a_{1,1} a_{2,4} a_{3,3} a_{4,2}-a_{1,3} a_{2,1} a_{3,4} a_{4,2}+a_{1,1} a_{2,3} a_{3,4} a_{4,2}+a_{1,4} a_{2,2} a_{3,1} a_{4,3}-a_{1,2} a_{2,4} a_{3,1} a_{4,3}-a_{1,4} a_{2,1} a_{3,2} a_{4,3}+a_{1,1} a_{2,4} a_{3,2} a_{4,3}+a_{1,2} a_{2,1} a_{3,4} a_{4,3}-a_{1,1} a_{2,2} a_{3,4} a_{4,3}-a_{1,3} a_{2,2} a_{3,1} a_{4,4}+a_{1,2} a_{2,3} a_{3,1} a_{4,4}+a_{1,3} a_{2,1} a_{3,2} a_{4,4}-a_{1,1} a_{2,3} a_{3,2} a_{4,4}-a_{1,2} a_{2,1} a_{3,3} a_{4,4}+a_{1,1} a_{2,2} a_{3,3} a_{4,4} \]

 

예

• 반데몬드 행렬 (Vandermonde matrix)

 

 

메모

• Háková, Lenka, and Agnieszka Tereszkiewicz. “On Immanant Functions Related to Weyl Groups of \(A_n\).” Journal of Mathematical Physics 55, no. 11 (November 2014): 113509. doi:10.1063/1.4901556.
• http://mathoverflow.net/questions/35988/why-were-matrix-determinants-once-such-a-big-deal
• 벡터의 스칼라 삼중곱\[\mathbf{a}\cdot(\mathbf{b}\times \mathbf{c})= \mathbf{b}\cdot(\mathbf{c}\times \mathbf{a})= \mathbf{c}\cdot(\mathbf{a}\times \mathbf{b}) = \begin{vmatrix} a_1 & a_2 & a_3 \\ b_1 & b_2 & b_3 \\ c_1 & c_2 & c_3 \\ \end{vmatrix}\]

관련된 항목들

• 행렬의 대각합 (trace)
• 벡터의 외적(cross product)
• 외대수(exterior algebra)와 겹선형대수(multilinear algebra)
• 파피안(Pfaffian)

 

 

수학용어번역

• 행렬식, determinant - 대한수학회 수학용어집
• parallelepiped - 대한수학회 수학용어집
  • 평행육면체?

 

 

매스매티카 파일 및 계산 리소스

• https://docs.google.com/file/d/0B8XXo8Tve1cxcE4yakhZTzBDYUE/edit
• http://stackoverflow.com/questions/tagged/determinants

사전 형태의 자료

• http://ko.wikipedia.org/wiki/행렬식
• http://en.wikipedia.org/wiki/Determinant
• http://en.wikipedia.org/wiki/Cauchy_matrix

리뷰, 에세이, 강의노트

• Abeles, Francine F. 2011. “Nineteenth Century Roots of Quasideterminants.” Linear Algebra and Its Applications 435 (5): 1019–1024. doi:10.1016/j.laa.2011.02.010.
• Krattenthaler, C. 2005. “Advanced Determinant Calculus: A Complement.” Linear Algebra and Its Applications 411: 68–166. doi:10.1016/j.laa.2005.06.042. http://arxiv.org/abs/math/0503507
• Krattenthaler, C. 1999. “Advanced Determinant Calculus.” Séminaire Lotharingien de Combinatoire 42: Art. B42q, 67 pp. (electronic). http://www.mat.univie.ac.at/~kratt/artikel/detsurv.html
• Brualdi, Richard A., and Hans Schneider. “Determinantal Identities: Gauss, Schur, Cauchy, Sylvester, Kronecker, Jacobi, Binet, Laplace, Muir, and Cayley.” Linear Algebra and Its Applications 52–53 (July 1983): 769–91. doi:10.1016/0024-3795(83)80049-4.

관련논문

• http://arxiv.org/abs/1512.08747
• Jing, Naihuan, and Jian Zhang. “Quantum Hyperdetermiants and Hyper-Pfaffians.” arXiv:1412.3612 [math], December 11, 2014. http://arxiv.org/abs/1412.3612.

노트

위키데이터

• ID : Q178546

말뭉치

1. The standard formula to find the determinant of a 3×3 matrix is a break down of smaller 2×2 determinant problems which are very easy to handle.
2. If you need a refresher, check out my other lesson on how to find the determinant of a 2×2.
3. In this section, we introduce the determinant of a matrix.
4. The symbol M ij represents the determinant of the matrix that results when row i and column j are eliminated.
5. To find the determinant of a 3 X 3 or larger matrix, first choose any row or column.
6. The sum of these products gives the value of the determinant.
7. But there is a condition to obtain a matrix determinant, the matrix must be a square matrix in order to calculate it.
8. Hence, the simplified definition is that the determinant is a value that can be computed from a square matrix to aid in the resolution of linear equation systems associated with such matrix.
9. The determinant of a matrix can be denoted simply as det A, det(A) or |A|.
10. This last notation comes from the notation we directly apply to the matrix we are obtaining the determinant of.
11. But, a determinant can be a negative number.
12. Besides, if the determinant of a matrix is non-zero, the linear system it represents is linearly independent.
13. In linear algebra, the determinant is a scalar value that can be computed from the elements of a square matrix and encodes certain properties of the linear transformation described by the matrix.
14. The determinant of a matrix A is denoted det(A), det A, or |A|.
15. This leads to the use of determinants in calculus, the Jacobian determinant in the change of variables rule for integrals of functions of several variables.
16. There are various equivalent ways to define the determinant of a square matrix A, i.e. one with the same number of rows and columns.
17. As a hint, I'll take the determinant of a very similar two by two matrix.
18. this is negative three... but making negative the negative three will make the positive three so the determinant of this matrix is twenty three.
19. Tool to compute a matrix determinant.
20. How to calculate a matrix determinant?
21. The determinant of a non-square matrix is not defined, it does not exist according to the definition of the determinant.
22. What is the formula for calculating the determinant of a matrix of order n?
23. As a hint, I will take the determinant of another 3 by 3 matrix.
24. But it's the exact same process for the 3 by 3 matrix that you're trying to find the determinant of.
25. And now let's evaluate its determinant.
26. So we could just write plus 4 times 4, the determinant of 4 submatrix.
27. A matrix is often used to represent the coefficients in a system of linear equations, and the determinant can be used to solve those equations.
28. The use of determinants in calculus includes the Jacobian determinant in the change of variables rule for integrals of functions of several variables.
29. It can be proven that any matrix has a unique inverse if its determinant is nonzero.
30. In linear algebra, the determinant is a value associated with a square matrix.
31. Notice the difference in notation between the matrix and its determinant: matrices are typically enclosed with square brackets whereas determinants of matrices are enclosed by straight lines.
32. This is called the expansion of the determinant by its first row.
33. Now take this determinant and multiply it by a (the element that was crossed out).
34. Now, the determinant is the sum of the products of the upper left to lower right diagonals minus the sum of the product of the upper right to lower left diagonals: |A|=(aek+bfg+cdh)-(bdk+afh+ceg).
35. You show that second matrix above as having a negative determinant.
36. To calculate the determinant of a matrix, you can choose any row or any column.
37. I only have to solve a determinant of order 3.
38. A determinant is a real number associated with every square matrix.
39. I have yet to find a good English definition for what a determinant is.
40. The determinant of a 2×2 matrix is found much like a pivot operation.
41. The determinant only exists for square matrices (2×2, 3×3, ... n×n).
42. The determinant of a 3 x 3 matrix is calculated for a matrix having 3 rows and 3 columns.
43. The symbol used to represent the determinant is represented by vertical lines on either side, such as | |.
44. Let A be the matrix, then the determinant of a matrix A is denoted by |A|.
45. We can find the determinant of a matrix in various ways.
46. The determinant will be equal to the product of that element and its cofactor.
47. Then, the determinant of is where in step we have used the fact that for all permutations except the product involves at least one entry above the main diagonal that is equal to zero.
48. We have proved above that matrices that have a zero row have zero determinant.
49. Thus, if is singular, and To sum up, we have proved that all invertible matrices have non-zero determinant, and all singular matrices have zero determinant.
50. If you take the values of one row and add them to a different row, the determinant of the matrix does not change.
51. The following examples illustrate the basic properties of the determinant of a matrix.
52. For \(2 \times 2\) matrices, the determinant is the area of the parallelogram defined by the rows (or columns), plotted in a 2D space.
53. There are many methods used for computing the determinant.
54. For a 2-by-2 matrix, the determinant is calculated by subtracting the reverse diagonal from the main diagonal, which is known as the Leibniz formula.
55. For more complicated matrices, the Laplace formula (cofactor expansion), Gaussian elimination or other algorithms must be used to calculate the determinant.
56. The determinant of any square matrix A is a scalar, denoted det(A).
57. The determinant function can be defined by essentially two different methods.
58. Method 1 for defining the determinant.
59. Now that the concepts of a permutation and its sign have been defined, the definition of the determinant of a matrix can be given.
60. In applying the definition to evaluate the determinant of a 7 by 7 matrix, for example, the sum (*) would contain more than five thousand terms.
61. The determinant is a unique number associated with a square matrix.
62. You would not want to calculate the determinant of a large matrix by hand.
63. Determinant of a Matrix is a special number that is defined only for square matrices (matrices which have same number of rows and columns).
64. In this Method We are using the properities of Determinant.
65. It uses the QR decomposition, a formula for the determinant of block diagonal matrices, a formula for the determinant of triangular matrices, and block multiplication of matrices.
66. The determinant of a square n × n matrix is calculated as the sum of n !
67. For the The determinant is a special scalar-valued function defined on the set of square matrices.
68. The term determinant was first introduced by the German mathematician Carl Friedrich Gauss in 1801.
69. There are various equivalent ways to define the determinant of a square matrix A , i.e., one with the same number of rows and columns.
70. A matrix determinant is difficult to define but a very useful number Unfortunately, not every square matrix has an inverse (although most do).
71. This scalar function of a square matrix is called the determinant.
72. The determinant of a matrix \({\bf A}\) is denoted by \(|{\bf A}|\).
73. As is the case of inversion of a square matrix, calculation of the determinant is tedious and computer assistance is needed for practical calculations.
74. As we said before, the idea is to assume that previous properties satisfied by the determinant of matrices of order 2, are still valid in general.
75. If we interchange two rows, the determinant of the new matrix is the opposite of the old one.
76. If we multiply one row with a constant, the determinant of the new matrix is the determinant of the old one multiplied by the constant.
77. If we add one row to another one multiplied by a constant, the determinant of the new matrix is the same as the old one.
78. In this section, we define the determinant, and we present one way to compute it.
79. We will give a recursive formula for the determinant in Section 4.2.
80. Scaling a row of A by a scalar c multiplies the determinant by c .
81. Swapping two rows of a matrix multiplies the determinant by − 1.
82. Note that the notation may be more convenient when indicating the absolute value of a determinant, i.e., instead of .