首先必须要判断矩阵是不是一个方阵。然后把在矩阵右边放一个单位矩阵，然后再进行行变换，左边变成单位矩阵后吗，右边的矩阵就是逆矩阵。
举个例子，以下矩阵：
[
1
1
1
1
2
1
1
1
−
1
2
1
1
−
3
2
1
2
]
left[begin{matrix} 1 & 1 & 1 & 1\ 2 & 1& 1& 1\ -1 & 2& 1& 1\ -3& 2& 1& 2\ end{matrix}right]
⎣⎢⎢⎡12−1−3112211111112⎦⎥⎥⎤
右接一个单位矩阵
[
1
1
1
1
1
0
0
0
2
1
1
1
0
1
0
0
−
1
2
1
1
0
0
1
0
−
3
2
1
2
0
0
0
1
]
left[begin{matrix} 1 & 1 & 1 & 1 & 1 & 0 & 0 & 0\ 2 & 1& 1& 1 & 0 & 1 & 0 & 0\ -1 & 2& 1& 1 & 0 & 0 & 1 & 0\ -3& 2& 1& 2 & 0 & 0 & 0 & 1\ end{matrix}right]
⎣⎢⎢⎡12−1−31122111111121000010000100001⎦⎥⎥⎤
进行行变换
[
1
1
1
1
1
0
0
0
0
1
1
1
2
−
1
0
0
0
0
−
1
−
1
−
5
3
1
0
0
0
0
1
−
2
2
−
1
1
]
left[begin{matrix} 1& 1 & 1 &1 & 1 & 0 & 0 &0\ 0& 1 & 1 &1& 2 &-1 & 0& 0\ 0 &0 &-1 &-1 &-5 &3 &1 &0\ 0 &0& 0& 1& -2& 2 &-1& 1\ end{matrix}right]
⎣⎢⎢⎡1000110011−1011−1112−5−20−132001−10001⎦⎥⎥⎤
单位化，就变成这样了：
[
1
0
0
0
−
1
1
0
0
0
1
0
0
−
3
2
1
0
0
0
1
0
7
−
5
0
−
1
0
0
0
1
−
2
2
−
1
1
]
left[begin{matrix} 1 & 0 & 0 & 0 & -1 & 1 & 0 & 0\ 0 & 1 & 0 & 0 & -3 & 2 & 1 & 0\ 0 & 0 & 1 & 0 & 7 & -5 & 0 & -1\ 0 & 0 & 0 & 1 & -2 & 2 & -1 & 1\ end{matrix}right]
⎣⎢⎢⎡1000010000100001−1−37−212−52010−100−11⎦⎥⎥⎤
Java代码：
public Matrixinverse() { // 必须是个方阵 if (this.array.length != this.array[0].length) { throw new IllegalArgumentException("方阵才有双侧逆矩阵"); } // 创建一个矩阵,单位矩阵放在右边 final T[][] newArray = newArray(this.array.length, this.array.length << 1); for (int i = 0; i < newArray.length; i++) { final T[] line = newArray[i]; final T[] thisLine = this.array[i]; for (int j = 0; j < thisLine.length; j++) { line[j] = thisLine[j]; } for (int j = thisLine.length; j < line.length; j++) { if (j == thisLine.length + i) { line[j] = onevalue(); } else { line[j] = zeroValue(); } } } return createMatrix(newArray).toUpperTriangular().toLowerTriangle().toUnipotent() .subMatrix(0, this.array.length, this.array.length, this.array.length * 2); }
python代码：
def __invert__(self):
# 首先新建一个单位矩阵
size = len(self.__lines)
unit = self.unit_matrix(size)
new_matrix = self.append(unit)
new_matrix.to_upper_triangle()
new_matrix.to_lower_triangle()
new_matrix.to_unipotent()
return new_matrix.sub_matrix(0, size, 0, size)
# 创建单位矩阵方法
# 简化后如下：
@staticmethod
def unit_matrix(size):
return [[1 if i == j else 0 for j in range(0, size)] for i in range(0, size)]
# 拼接矩阵方法
def append(self, matrix):
# define an array
rows = len(self.__lines)
columns = len(self.__lines[0]) + len(matrix.__lines[0])
unit = [[0 for _ in range(0, columns)] for _ in range(0, rows)]
for y in range(0, len(unit)):
self_line = self.__lines[y]
other_line = matrix.__lines[y]
# 0~ this this ~other
for i in range(0, len(self_line)):
unit[y][i] = self_line[i]
for i in range(0, len(other_line)):
unit[y][i+len(self_line)] = other_line[i]
return Matrix(unit)
# 下三角矩阵方法
def to_lower_triangle(self):
for i in range(len(self.__lines) - 1, -1, -1):
# 循环遍历其前的所有行, 其前所有行进行改变
for j in range(0, i):
Matrix.row_operate(self.__lines[i], self.__lines[j], self.__lines[j][i], self.__lines[i][i],
0, len(self.__lines[i]))
# 单位矩阵方法
def to_unipotent(self):
for i in range(0, len(self.__lines)):
line = self.__lines[i]
for j in range(0, len(line)):
if i != j and line[i] != 0:
line[j] /= line[i]
# 行变换方法
@staticmethod
def __row_operate(line1, line2, coefficient1, coefficient2, start_column, end_column):
for i in range(start_column, end_column):
line2[i] = line1[i] * coefficient1 - line2[i] * coefficient2
