Matrix of Minors

matrix_of_minors(matrix)

Create the matrix of minors corresponding to a given matrix

Parameters

matrix (list of lists of int or float) – List of lists of numbers representing a matrix

Raises

  • TypeError – Argument must be a 2-dimensional list

  • TypeError – Elements nested within argument must be integers or floats

  • ValueError – Argument must contain the same amount of lists as the amount of elements contained within its first list

Returns

matrix – List of lists in which each inner element is a determinant of a subsection of the original matrix

Return type

list of lists of float

See also

matrix_of_cofactors(), transposed_matrix(), linear_determinant(), inverse_matrix()

Notes

  • Original matrix: \(\mathbf{A} = \begin{bmatrix} a_{1,1} & a_{1,2} & \cdots & a_{1,j} & a_{1,n} \\ a_{2,1} & a_{2,2} & \cdots & a_{2,j} & a_{2,n} \\ a_{i,1} & a_{i,2} & \cdots & a_{i,j} & a_{i,n} \\ \cdots & \cdots & \cdots & \cdots & \cdots \\ a_{m,1} & a_{m,2} & \cdots & a_{m,j} & a_{m,n} \end{bmatrix}\)

  • Minor of matrix: \(|\mathbf{A}_{i,j}| = \begin{vmatrix} a_{1,1} & a_{1,2} & \cdots & a_{1,n} \\ a_{2,1} & a_{2,2} & \cdots & a_{2,n} \\ \cdots & \cdots & \cdots & \cdots \\ a_{m,1} & a_{m,2} & \cdots & a_{m,n} \end{vmatrix}\)

  • Matrix of minors: \(\mathbf{A}^M = \begin{bmatrix} |\mathbf{A}_{1,1}| & |\mathbf{A}_{1,2}| & \cdots & |\mathbf{A}_{1,j}| & |\mathbf{A}_{1,n}| \\ |\mathbf{A}_{2,1}| & |\mathbf{A}_{2,2}| & \cdots & |\mathbf{A}_{2,j}| & |\mathbf{A}_{2,n}| \\ |\mathbf{A}_{i,1}| & |\mathbf{A}_{i,2}| & \cdots & |\mathbf{A}_{i,j}| & |\mathbf{A}_{i,n}| \\ \cdots & \cdots & \cdots & \cdots & \cdots \\ |\mathbf{A}_{m,1}| & |\mathbf{A}_{m,2}| & \cdots & |\mathbf{A}_{m,j}| & |\mathbf{A}_{m,n}| \end{bmatrix}\)

  • Matrix of Minors

Examples

Import matrix_of_minors function from regressions library
>>> from regressions.matrices.minors import matrix_of_minors
Create the matrix of minors for [[1, 2], [3, 4]]
>>> matrix_2x2 = matrix_of_minors([[1, 2], [3, 4]])
>>> print(matrix_2x2)
[[4.0, 3.0], [2.0, 1.0]]
Create the matrix of minors for [[2, 3, 5], [7, 11, 13], [17, 19, 23]]
>>> matrix_3x3 = matrix_of_minors([[2, 3, 5], [7, 11, 13], [17, 19, 23]])
>>> print(matrix_3x3)
[[6.0, -60.0, -54.0], [-26.0, -39.0, -13.0], [-16.0, -9.0, 1.0]]