가우스-요르단 소거법

개요

• 선형대수학의 중요한 알고리즘의 하나
• 선형연립방정식의 해법, 역행렬의 계산 등에 활용할 수 있다

 

 

예

\(\left( \begin{array}{ccc} 1 & -3 & 0 \\ -1 & 1 & 5 \\ 0 & 1 & 1 \end{array} \right)\) 에 가우스-조단 소거법을 적용한 경우

 

\(\begin{array}{l} \left( \begin{array}{ccc} 1 & -3 & 0 \\ -1 & 1 & 5 \\ 0 & 1 & 1 \end{array} \right) \\ \left( \begin{array}{ccc} 1 & -3 & 0 \\ 0 & -2 & 5 \\ 0 & 1 & 1 \end{array} \right) \\ \left( \begin{array}{ccc} 1 & -3 & 0 \\ 0 & 1 & -\frac{5}{2} \\ 0 & 1 & 1 \end{array} \right) \\ \left( \begin{array}{ccc} 1 & 0 & -\frac{15}{2} \\ 0 & 1 & -\frac{5}{2} \\ 0 & 1 & 1 \end{array} \right) \\ \left( \begin{array}{ccc} 1 & 0 & -\frac{15}{2} \\ 0 & 1 & -\frac{5}{2} \\ 0 & 0 & \frac{7}{2} \end{array} \right) \\ \left( \begin{array}{ccc} 1 & 0 & -\frac{15}{2} \\ 0 & 1 & -\frac{5}{2} \\ 0 & 0 & 1 \end{array} \right) \\ \left( \begin{array}{ccc} 1 & 0 & 0 \\ 0 & 1 & -\frac{5}{2} \\ 0 & 0 & 1 \end{array} \right) \\ \left( \begin{array}{ccc} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{array} \right) \end{array}\)

 

 

메모

• http://math.fullerton.edu/mathews/n2003/GaussianJordanMod.html

관련된 항목들

• 가우스 소거법

매스매티카 파일 및 계산 리소스

• https://docs.google.com/file/d/0B8XXo8Tve1cxY2xCTnByU2hWZDg/edit

노트

위키데이터

• ID : Q1195020

말뭉치

1. Here you can solve systems of simultaneous linear equations using Gauss-Jordan Elimination Calculator with complex numbers online for free with a very detailed solution.
2. In fact Gauss-Jordan elimination algorithm is divided into forward elimination and back substitution.
3. But practically it is more convenient to eliminate all elements below and above at once when using Gauss-Jordan elimination calculator.
4. To understand Gauss-Jordan elimination algorithm better input any example, choose "very detailed solution" option and examine the solution.
5. Gauss-Jordan Elimination is an algorithm that can be used to solve systems of linear equations and to find the inverse of any invertible matrix.
6. It's called Gauss-Jordan elimination, to find the inverse of the matrix.
7. And we did this using Gauss-Jordan elimination.
8. To obtain an initial basic feasible solution, the Gauss-Jordan elimination procedure can be used to convert the Ax = b in the canonical form.
9. Using row operations to convert a matrix into reduced row echelon form is sometimes called Gauss–Jordan elimination.
10. A variant of Gaussian elimination called Gauss–Jordan elimination can be used for finding the inverse of a matrix, if it exists.
11. To convert any matrix to its reduced row echelon form, Gauss-Jordan elimination is performed.
12. Gauss-Jordan elimination is a mechanical procedure for transforming a given system of linear equations to \(Rx = d\) with \(R\) in RREF using only elementary row operations.
13. This more-complete method of solving is called "Gauss-Jordan elimination" (with the equations ending up in what is called "reduced-row-echelon form").
14. Gauss–Jordan elimination is backward stable for matrices diagonally dominant by rows and not for those diagonally dominant by columns.
15. Use Gauss-Jordan elimination on augmented matrices to solve a linear system and calculate the matrix inverse.
16. We apply the Gauss-Jordan Elimination method: we obtain the reduced row echelon form from the augmented matrix of the equation system by performing elemental operations in rows (or columns).
17. But in case of Gauss-Jordan Elimination Method, we only have to form a reduced row echelon form (diagonal matrix).
18. Gauss-Jordan Elimination Method can be used for finding the solution of a systems of linear equations which is applied throughout the mathematics.
19. The Gauss-Jordan Elimination method can be used in determining the inverse of a square matrix.