本项目为关于矩阵分解的LU、QR(Gram-Schmidt)、Orthogonal Reduction (Householder reduction 和Givens reduction)和URV程序实现。并包含一个综合程序，根据选择参数的不同，实现不同的矩阵分解；并实现Ax=b方程组的求解，以及计算A的行列式。
- python 3.8
- numpy 1.23.4
main.py # 主程序
MatrixExample.py # 存储了一些矩阵样本
LUFactorization.py # LU 分解
QRFactorization.py # QR(Gram-Schmidt) 分解
HousholderReduction.py # Housholder 约简
GivensReduction.py # Givens 约简
URVFactorization.py # URV 分解
配置好环境以后，直接运行py文件。
主程序运行后会进入循环，首先根据提示输入1-5选择分解方法，输入6退出程序；其次输入1-7选择矩阵，矩阵在MatrixExample.py文件中定义。然后程序会计算矩阵的行列式以及使用选定分解方法的结果。
在预设的矩阵中，1-2包含A和b矩阵，会返回A的分解结果和Ax=b的解；3-7只包含A矩阵，不包含b矩阵，故只返回A的分解结果
注意：本程序中的Ax=b仅支持含有唯一解的情况，即rank(A)=rank(A|b)
> python .\main.py
----------------------------------------------
Please choose one method:
1. LU Factorization
2. QR Factorization
3. Householder Reduction
4. Givens Reduction
5. URV Reduction
6. Exit the program!
2
Please type in the index (1-7) of the matrix defined in the MatrixExample.py!
2
The determinant of the input matrix is: 25.0
The input matrix A is:
[[ 2. -3. 1.]
[ 1. -1. -2.]
[ 3. 1. -1.]]
The input matrix b is:
[ 7. -2. 0.]
The input matrix is:
[[ 2. -3. 1.]
[ 1. -1. -2.]
[ 3. 1. -1.]]
The result of the QR factorization is shown as follows:
-----The Q matrix is:
[[ 0.535 -0.774 0.341]
[ 0.267 -0.228 -0.936]
[ 0.802 0.592 0.085]]
-----The R matrix is:
[[ 3.742 -1.069 -0.802]
[ 0. 3.14 -0.91 ]
[-0. -0. 2.128]]
The solution of Qy = b is:
[ 3.207 -4.96 4.256]
The solution of Rx = y is:
[[ 1.]
[-1.]
[ 2.]]
LU分解要求被分解矩阵为方阵，且为非奇异矩阵。测试样例中，包含两个3x3非奇异矩阵、一个3x4矩阵和一个3x3奇异矩阵
> python .\LUFactorization.py
结果如下：
The input matrix is:
[[ 1. 2. 5.]
[ 4. 20. 6.]
[ 7. 1. 9.]]
The result of the LU factorization is shown as follows:
-----The L matrix is:
[[1. 0. 0. ]
[0.571 1. 0. ]
[0.143 0.096 1. ]]
-----The U matrix is:
[[ 7. 1. 9. ]
[ 0. 19.429 0.857]
[ 0. 0. 3.632]]
-----The P matrix is:
[[0. 0. 1.]
[0. 1. 0.]
[1. 0. 0.]]
----------------------------------------------
The input matrix is:
[[10. 2. 4.]
[ 6. 5. 2.]
[ 7. 15. 9.]]
The result of the LU factorization is shown as follows:
-----The L matrix is:
[[1. 0. 0. ]
[0.7 1. 0. ]
[0.6 0.279 1. ]]
-----The U matrix is:
[[10. 2. 4. ]
[ 0. 13.6 6.2 ]
[ 0. -0. -2.132]]
-----The P matrix is:
[[1. 0. 0.]
[0. 0. 1.]
[0. 1. 0.]]
----------------------------------------------
The input matrix is:
[[10. 2. 4. 8.]
[ 6. 5. 2. 1.]
[ 7. 15. 9. 6.]]
ERROR: The input matrix is not a square matrix.
----------------------------------------------
The input matrix is:
[[10. 2. 4.]
[10. 2. 4.]
[ 7. 15. 9.]]
ERROR: The matrix is not invertible.
QR分解要求被分解矩阵为列向量无关的mxn矩阵，测试样例中，包含两个3x3非奇异矩阵、一个3x4矩阵和一个3x3奇异矩阵
> python .\QRFactorization.py
结果如下：
The input matrix is:
[[ 1. 2. 5.]
[ 4. 20. 6.]
[ 7. 1. 9.]]
The result of the QR factorization is shown as follows:
-----The Q matrix is:
[[ 0.123 0.039 0.992]
[ 0.492 0.865 -0.095]
[ 0.862 -0.5 -0.087]]
-----The R matrix is:
[[ 8.124 10.955 11.324]
[ 0. 16.881 0.885]
[-0. -0. 3.602]]
----------------------------------------------
The input matrix is:
[[10. 2. 4.]
[ 6. 5. 2.]
[ 7. 15. 9.]]
The result of the QR factorization is shown as follows:
-----The Q matrix is:
[[ 0.735 -0.572 0.363]
[ 0.441 -0.002 -0.897]
[ 0.515 0.82 0.251]]
-----The R matrix is:
[[13.601 11.396 8.455]
[ 0. 11.142 5.084]
[ 0. 0. 1.914]]
----------------------------------------------
The input matrix is:
[[10. 2. 4. 8.]
[ 6. 5. 2. 1.]
[ 7. 15. 9. 6.]]
ERROR: The column vertor is not linearly independent.
----------------------------------------------
The input matrix is:
[[10. 2. 4.]
[10. 2. 4.]
[ 7. 15. 9.]]
ERROR: The column vertor is not linearly independent.
Housholder 约简的简单实现，测试样例中，包含两个3*3非奇异矩阵、一个3x3奇异矩阵、一个3x4矩阵和一个4x3矩阵
> python .\HousholderReduction.py
结果如下：
The input matrix is:
[[1. 2. 3.]
[4. 5. 6.]
[7. 8. 9.]]
The result of the Housholder Reduction is shown as follows:
-----The P matrix is:
[[ 0.123 0.492 0.862]
[ 0.905 0.302 -0.302]
[-0.408 0.816 -0.408]]
-----The T matrix is:
[[ 8.124 9.601 11.078]
[-0. 0.905 1.809]
[ 0. 0. 0. ]]
----------------------------------------------
The input matrix is:
[[10. 2. 4.]
[ 6. 5. 2.]
[ 7. 15. 9.]]
The result of the Housholder Reduction is shown as follows:
-----The P matrix is:
[[ 0.735 0.441 0.515]
[-0.572 -0.002 0.82 ]
[ 0.363 -0.897 0.251]]
-----The T matrix is:
[[13.601 11.396 8.455]
[-0. 11.142 5.084]
[ 0. -0. 1.914]]
----------------------------------------------
The input matrix is:
[[10. 2. 4.]
[10. 2. 4.]
[ 7. 15. 9.]]
The result of the Housholder Reduction is shown as follows:
-----The P matrix is:
[[ 0.634 0.634 0.444]
[-0.314 -0.314 0.896]
[ 0.707 -0.707 0. ]]
-----The T matrix is:
[[15.78 9.189 9.062]
[ 0. 12.189 5.557]
[-0. 0. -0. ]]
----------------------------------------------
The input matrix is:
[[10. 2. 4. 8.]
[ 6. 5. 2. 1.]
[ 7. 15. 9. 6.]]
The result of the Housholder Reduction is shown as follows:
-----The P matrix is:
[[ 0.735 0.441 0.515]
[-0.572 -0.002 0.82 ]
[ 0.363 -0.897 0.251]]
-----The T matrix is:
[[13.601 11.396 8.455 9.411]
[-0. 11.142 5.084 0.337]
[ 0. -0. 1.914 3.511]]
----------------------------------------------
The input matrix is:
[[10. 6. 7.]
[ 2. 5. 15.]
[ 4. 2. 9.]
[ 8. 1. 6.]
[10. 2. 4.]]
The result of the Housholder Reduction is shown as follows:
-----The P matrix is:
[[ 0.593 0.119 0.237 0.475 0.593]
[ 0.411 0.771 0.092 -0.36 -0.314]
[-0.608 0.452 0.502 0.418 -0.017]
[ 0.127 0.116 -0.531 0.677 -0.48 ]
[ 0.306 -0.417 0.634 0.111 -0.564]]
-----The T matrix is:
[[16.852 6.29 13.292]
[ 0. 5.517 11.853]
[-0. -0. 9.477]
[ 0. -0. -0. ]
[ 0. 0. 0. ]]
Givens 约简的简单实现，测试样例中，包含两个3*3非奇异矩阵、一个3x3奇异矩阵、一个3x4矩阵和一个4x3矩阵
> python .\GivensReduction.py
结果如下：
The input matrix is:
[[1. 2. 3.]
[4. 5. 6.]
[7. 8. 9.]]
3
The result of the Givens Reduction is shown as follows:
-----The P matrix is:
[[ 0.123 0.492 0.862]
[ 0.905 0.302 -0.302]
[-0.408 0.816 -0.408]]
-----The T matrix is:
[[ 8.124 9.601 11.078]
[-0. 0.905 1.809]
[ 0. -0. 0. ]]
----------------------------------------------
The input matrix is:
[[10. 2. 4.]
[ 6. 5. 2.]
[ 7. 15. 9.]]
3
The result of the Givens Reduction is shown as follows:
-----The P matrix is:
[[ 0.735 0.441 0.515]
[-0.572 -0.002 0.82 ]
[ 0.363 -0.897 0.251]]
-----The T matrix is:
[[13.601 11.396 8.455]
[ 0. 11.142 5.084]
[-0. 0. 1.914]]
----------------------------------------------
The input matrix is:
[[10. 2. 4.]
[10. 2. 4.]
[ 7. 15. 9.]]
3
The result of the Givens Reduction is shown as follows:
-----The P matrix is:
[[ 0.634 0.634 0.444]
[-0.314 -0.314 0.896]
[ 0.707 -0.707 -0. ]]
-----The T matrix is:
[[15.78 9.189 9.062]
[-0. 12.189 5.557]
[-0. -0. -0. ]]
----------------------------------------------
The input matrix is:
[[10. 2. 4. 8.]
[ 6. 5. 2. 1.]
[ 7. 15. 9. 6.]]
4
The result of the Givens Reduction is shown as follows:
-----The P matrix is:
[[ 0.735 0.441 0.515]
[-0.572 -0.002 0.82 ]
[ 0.363 -0.897 0.251]]
-----The T matrix is:
[[13.601 11.396 8.455 9.411]
[ 0. 11.142 5.084 0.337]
[-0. 0. 1.914 3.511]]
----------------------------------------------
The input matrix is:
[[10. 6. 7.]
[ 2. 5. 15.]
[ 4. 2. 9.]
[ 8. 1. 6.]]
3
The result of the Givens Reduction is shown as follows:
-----The P matrix is:
[[ 0.737 0.147 0.295 0.59 ]
[ 0.261 0.8 0.026 -0.539]
[-0.615 0.458 0.498 0.405]
[-0.101 0.358 -0.815 0.444]]
-----The T matrix is:
[[13.565 6.34 13.565]
[-0. 5.08 10.827]
[-0. -0. 9.475]
URV 分解的简单实现，测试样例中，包含两个3*3非奇异矩阵
> python .\URVFactorization.py
结果如下：
The input matrix is:
[[1. 2. 3.]
[4. 5. 6.]
[7. 8. 9.]]
The result of the URV Factorization is shown as follows:
-----The U matrix is:
[[ 0.123 0.905 -0.408]
[ 0.492 0.302 0.816]
[ 0.862 -0.302 -0.408]]
-----The R matrix is:
[[16.76 -0. 0. ]
[ 1.714 1.074 -0. ]
[ 0. 0. 0. ]]
-----The V matrix is:
[[ 0.485 0.573 0.661]
[-0.774 -0.072 0.63 ]
[ 0.408 -0.816 0.408]]
----------------------------------------------
The input matrix is:
[[10. 2. 4.]
[10. 2. 4.]
[ 7. 15. 9.]]
The result of the URV Factorization is shown as follows:
-----The U matrix is:
[[ 0.634 -0.314 0.707]
[ 0.634 -0.314 -0.707]
[ 0.444 0.896 0. ]]
-----The R matrix is:
[[20.385 0. 0. ]
[ 7.964 10.771 0. ]
[-0. 0. -0. ]]
-----The V matrix is:
[[ 0.774 0.451 0.445]
[-0.572 0.798 0.187]
[-0.271 -0.399 0.876]]