|
| 1 | +"""Determine whether two square matrices are similar. |
| 2 | +
|
| 3 | +Two square matrices :math:`A` and :math:`B` of the same size are similar if |
| 4 | +there exists an invertible matrix :math:`P` such that :math:`P^{-1} A P = B`. |
| 5 | +This implementation relies on SymPy to compute the Jordan canonical form of |
| 6 | +both matrices. Two matrices are similar precisely when their Jordan forms are |
| 7 | +equal up to permutation of the Jordan blocks. |
| 8 | +* https://en.wikipedia.org/wiki/Jordan_matrix |
| 9 | +* https://en.wikipedia.org/wiki/Jordan_normal_form |
| 10 | +
|
| 11 | +Examples |
| 12 | +-------- |
| 13 | +>>> are_similar_matrices([[3, 1], [0, 3]], [[3, 0], [0, 3]]) |
| 14 | +False |
| 15 | +>>> from sympy import Matrix |
| 16 | +>>> matrix_a = [[3, 1], [0, 3]] |
| 17 | +>>> transform = Matrix([[1, 1], [0, 1]]) |
| 18 | +>>> matrix_b = (transform.inv() * Matrix(matrix_a) * transform).tolist() |
| 19 | +>>> are_similar_matrices(matrix_a, matrix_b) |
| 20 | +True |
| 21 | +>>> are_similar_matrices( |
| 22 | +... [[1, 2, 0], [0, 1, 0], [0, 0, 3]], |
| 23 | +... [[1, 0, 0], [0, 1, 0], [0, 0, 3]], |
| 24 | +... ) |
| 25 | +False |
| 26 | +>>> are_similar_matrices([[1, 2], [0, 1]], [[1, 2, 0], [0, 1, 0]]) |
| 27 | +Traceback (most recent call last): |
| 28 | + ... |
| 29 | +ValueError: both matrices must be square with the same dimensions |
| 30 | +""" |
| 31 | + |
| 32 | +from __future__ import annotations |
| 33 | + |
| 34 | +from collections.abc import Sequence |
| 35 | +from typing import Any |
| 36 | + |
| 37 | +from sympy import Matrix, nsimplify |
| 38 | +from sympy.matrices.common import MatrixError |
| 39 | + |
| 40 | +__all__ = ["are_similar_matrices"] |
| 41 | + |
| 42 | + |
| 43 | +type MatrixLike = Sequence[Sequence[Any]] | Matrix |
| 44 | + |
| 45 | + |
| 46 | +def _as_square_matrix(matrix: MatrixLike, *, simplify_entries: bool) -> Matrix: |
| 47 | + """Return a SymPy matrix after validating that ``matrix`` is square. |
| 48 | +
|
| 49 | + Parameters |
| 50 | + ---------- |
| 51 | + matrix: |
| 52 | + Nested sequences (or a SymPy matrix) describing the matrix entries. |
| 53 | + simplify_entries: |
| 54 | + When ``True`` each entry is passed through :func:`sympy.nsimplify` |
| 55 | + which helps treat values such as ``0.5`` and ``1/2`` as the same. |
| 56 | +
|
| 57 | + Raises |
| 58 | + ------ |
| 59 | + TypeError |
| 60 | + If ``matrix`` cannot be converted into a SymPy matrix. |
| 61 | + ValueError |
| 62 | + If ``matrix`` is not square. |
| 63 | + """ |
| 64 | + |
| 65 | + try: |
| 66 | + sympy_matrix = Matrix(matrix) |
| 67 | + except (TypeError, ValueError) as exc: # pragma: no cover - defensive |
| 68 | + msg = "matrix input must be a rectangular sequence of numbers" |
| 69 | + raise TypeError(msg) from exc |
| 70 | + |
| 71 | + if sympy_matrix.rows != sympy_matrix.cols: |
| 72 | + raise ValueError("both matrices must be square with the same dimensions") |
| 73 | + |
| 74 | + if simplify_entries: |
| 75 | + sympy_matrix = sympy_matrix.applyfunc(nsimplify) |
| 76 | + |
| 77 | + return sympy_matrix |
| 78 | + |
| 79 | + |
| 80 | +def _jordan_signature(matrix: Matrix) -> tuple[tuple[Any, tuple[int, ...]], ...]: |
| 81 | + """Return a hashable representation of the Jordan form of ``matrix``.""" |
| 82 | + |
| 83 | + _, blocks = matrix.jordan_cells() |
| 84 | + summary: dict[Any, list[int]] = {} |
| 85 | + for block in blocks: |
| 86 | + block_matrix = Matrix(block) |
| 87 | + eigenvalue = block_matrix[0, 0] |
| 88 | + summary.setdefault(eigenvalue, []).append(block_matrix.rows) |
| 89 | + |
| 90 | + return tuple( |
| 91 | + ( |
| 92 | + eigenvalue, |
| 93 | + tuple(sorted(block_sizes, reverse=True)), |
| 94 | + ) |
| 95 | + for eigenvalue, block_sizes in sorted( |
| 96 | + summary.items(), key=lambda item: repr(item[0]) |
| 97 | + ) |
| 98 | + ) |
| 99 | + |
| 100 | + |
| 101 | +def are_similar_matrices( |
| 102 | + matrix_a: MatrixLike, |
| 103 | + matrix_b: MatrixLike, |
| 104 | + *, |
| 105 | + simplify_entries: bool = True, |
| 106 | +) -> bool: |
| 107 | + """Return ``True`` if ``matrix_a`` and ``matrix_b`` are similar matrices. |
| 108 | +
|
| 109 | + Parameters |
| 110 | + ---------- |
| 111 | + matrix_a, matrix_b: |
| 112 | + Square matrices represented as nested sequences (or SymPy matrices). |
| 113 | + simplify_entries: |
| 114 | + If ``True`` (default), the function attempts to simplify each entry so |
| 115 | + that values that are algebraically equal are treated as such. Set this |
| 116 | + to ``False`` to skip simplification when working with symbolic inputs |
| 117 | + that should remain untouched. |
| 118 | +
|
| 119 | + Raises |
| 120 | + ------ |
| 121 | + ValueError |
| 122 | + If the matrices are not square or their dimensions do not match. |
| 123 | + TypeError |
| 124 | + If either matrix cannot be interpreted as a numeric matrix. |
| 125 | + """ |
| 126 | + |
| 127 | + sympy_a = _as_square_matrix(matrix_a, simplify_entries=simplify_entries) |
| 128 | + sympy_b = _as_square_matrix(matrix_b, simplify_entries=simplify_entries) |
| 129 | + |
| 130 | + if sympy_a.shape != sympy_b.shape: |
| 131 | + raise ValueError("both matrices must be square with the same dimensions") |
| 132 | + |
| 133 | + try: |
| 134 | + signature_a = _jordan_signature(sympy_a) |
| 135 | + signature_b = _jordan_signature(sympy_b) |
| 136 | + except MatrixError as exc: # pragma: no cover - rare SymPy failure |
| 137 | + msg = "unable to determine the Jordan canonical form" |
| 138 | + raise ValueError(msg) from exc |
| 139 | + |
| 140 | + return signature_a == signature_b |
| 141 | + |
| 142 | + |
| 143 | +if __name__ == "__main__": # pragma: no cover - convenience execution |
| 144 | + from doctest import testmod |
| 145 | + |
| 146 | + testmod() |
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