Add similarity test for square matrices (#13022)

* Add similarity test for square matrices

* Fix naming convention: MatrixLike to matrix_like

* Use TypeAlias for proper mypy type checking

* Fix CI errors: use type keyword and fix line length

* Fix docstring formatting in similar_matrices.py

Corrected formatting and punctuation in docstring.

* updating DIRECTORY.md

---------

Co-authored-by: Christian Clauss <cclauss@me.com>
Co-authored-by: cclauss <cclauss@users.noreply.github.com>

Author: PixelCode01

DIRECTORY.md

@@ -1028,6 +1028,7 @@
   * [Rotate Matrix](matrix/rotate_matrix.py)
   * [Searching In Sorted Matrix](matrix/searching_in_sorted_matrix.py)
   * [Sherman Morrison](matrix/sherman_morrison.py)
+  * [Similar Matrices](matrix/similar_matrices.py)
   * [Spiral Print](matrix/spiral_print.py)
   * Tests
     * [Test Matrix Operation](matrix/tests/test_matrix_operation.py)

matrix/similar_matrices.py

@@ -0,0 +1,146 @@
+"""Determine whether two square matrices are similar.
+
+Two square matrices :math:`A` and :math:`B` of the same size are similar if
+there exists an invertible matrix :math:`P` such that :math:`P^{-1} A P = B`.
+This implementation relies on SymPy to compute the Jordan canonical form of
+both matrices.  Two matrices are similar precisely when their Jordan forms are
+equal up to permutation of the Jordan blocks.
+* https://en.wikipedia.org/wiki/Jordan_matrix
+* https://en.wikipedia.org/wiki/Jordan_normal_form
+
+Examples
+--------
+>>> are_similar_matrices([[3, 1], [0, 3]], [[3, 0], [0, 3]])
+False
+>>> from sympy import Matrix
+>>> matrix_a = [[3, 1], [0, 3]]
+>>> transform = Matrix([[1, 1], [0, 1]])
+>>> matrix_b = (transform.inv() * Matrix(matrix_a) * transform).tolist()
+>>> are_similar_matrices(matrix_a, matrix_b)
+True
+>>> are_similar_matrices(
+...     [[1, 2, 0], [0, 1, 0], [0, 0, 3]],
+...     [[1, 0, 0], [0, 1, 0], [0, 0, 3]],
+... )
+False
+>>> are_similar_matrices([[1, 2], [0, 1]], [[1, 2, 0], [0, 1, 0]])
+Traceback (most recent call last):
+    ...
+ValueError: both matrices must be square with the same dimensions
+"""
+
+from __future__ import annotations
+
+from collections.abc import Sequence
+from typing import Any
+
+from sympy import Matrix, nsimplify
+from sympy.matrices.common import MatrixError
+
+__all__ = ["are_similar_matrices"]
+
+
+type MatrixLike = Sequence[Sequence[Any]] | Matrix
+
+
+def _as_square_matrix(matrix: MatrixLike, *, simplify_entries: bool) -> Matrix:
+    """Return a SymPy matrix after validating that ``matrix`` is square.
+
+    Parameters
+    ----------
+    matrix:
+        Nested sequences (or a SymPy matrix) describing the matrix entries.
+    simplify_entries:
+        When ``True`` each entry is passed through :func:`sympy.nsimplify`
+        which helps treat values such as ``0.5`` and ``1/2`` as the same.
+
+    Raises
+    ------
+    TypeError
+        If ``matrix`` cannot be converted into a SymPy matrix.
+    ValueError
+        If ``matrix`` is not square.
+    """
+
+    try:
+        sympy_matrix = Matrix(matrix)
+    except (TypeError, ValueError) as exc:  # pragma: no cover - defensive
+        msg = "matrix input must be a rectangular sequence of numbers"
+        raise TypeError(msg) from exc
+
+    if sympy_matrix.rows != sympy_matrix.cols:
+        raise ValueError("both matrices must be square with the same dimensions")
+
+    if simplify_entries:
+        sympy_matrix = sympy_matrix.applyfunc(nsimplify)
+
+    return sympy_matrix
+
+
+def _jordan_signature(matrix: Matrix) -> tuple[tuple[Any, tuple[int, ...]], ...]:
+    """Return a hashable representation of the Jordan form of ``matrix``."""
+
+    _, blocks = matrix.jordan_cells()
+    summary: dict[Any, list[int]] = {}
+    for block in blocks:
+        block_matrix = Matrix(block)
+        eigenvalue = block_matrix[0, 0]
+        summary.setdefault(eigenvalue, []).append(block_matrix.rows)
+
+    return tuple(
+        (
+            eigenvalue,
+            tuple(sorted(block_sizes, reverse=True)),
+        )
+        for eigenvalue, block_sizes in sorted(
+            summary.items(), key=lambda item: repr(item[0])
+        )
+    )
+
+
+def are_similar_matrices(
+    matrix_a: MatrixLike,
+    matrix_b: MatrixLike,
+    *,
+    simplify_entries: bool = True,
+) -> bool:
+    """Return ``True`` if ``matrix_a`` and ``matrix_b`` are similar matrices.
+
+    Parameters
+    ----------
+    matrix_a, matrix_b:
+        Square matrices represented as nested sequences (or SymPy matrices).
+    simplify_entries:
+        If ``True`` (default), the function attempts to simplify each entry so
+        that values that are algebraically equal are treated as such. Set this
+        to ``False`` to skip simplification when working with symbolic inputs
+        that should remain untouched.
+
+    Raises
+    ------
+    ValueError
+        If the matrices are not square or their dimensions do not match.
+    TypeError
+        If either matrix cannot be interpreted as a numeric matrix.
+    """
+
+    sympy_a = _as_square_matrix(matrix_a, simplify_entries=simplify_entries)
+    sympy_b = _as_square_matrix(matrix_b, simplify_entries=simplify_entries)
+
+    if sympy_a.shape != sympy_b.shape:
+        raise ValueError("both matrices must be square with the same dimensions")
+
+    try:
+        signature_a = _jordan_signature(sympy_a)
+        signature_b = _jordan_signature(sympy_b)
+    except MatrixError as exc:  # pragma: no cover - rare SymPy failure
+        msg = "unable to determine the Jordan canonical form"
+        raise ValueError(msg) from exc
+
+    return signature_a == signature_b
+
+
+if __name__ == "__main__":  # pragma: no cover - convenience execution
+    from doctest import testmod
+
+    testmod()