POLAR

The polar decomposition factors a matrix A into the product of an orthonormal matrix U and a positive semidefinite Hermitian matrix P.

Excel Usage

=POLAR(matrix, polar_side, polar_ret_type)

matrix (list[list], required): 2D array of numeric values to decompose.
polar_side (str, optional, default: “right”): Determines whether a right (A=UP) or left (A=PU) polar decomposition is computed.
polar_ret_type (str, optional, default: “u”): The component of the decomposition to return.

Returns (list[list]): 2D array representing the requested component (U or P).

Example 1: Unitary matrix U of 2x2

Inputs:

matrix		polar_ret_type
1	-1	u
2	4

Excel formula:

=POLAR({1,-1;2,4}, "u")

Expected output:

Result
0.857493	-0.514496
0.514496	0.857493

Example 2: Positive semidefinite matrix P of 2x2

Inputs:

matrix		polar_ret_type
1	-1	p
2	4

Excel formula:

=POLAR({1,-1;2,4}, "p")

Expected output:

Result
1.88648	1.20049
1.20049	3.94447

Python Code

import numpy as np
from scipy.linalg import polar as scipy_polar

def polar(matrix, polar_side='right', polar_ret_type='u'):
    """
    Compute the polar decomposition of a matrix.

    See: https://docs.scipy.org/doc/scipy/reference/generated/scipy.linalg.polar.html

    This example function is provided as-is without any representation of accuracy.

    Args:
        matrix (list[list]): 2D array of numeric values to decompose.
        polar_side (str, optional): Determines whether a right (A=UP) or left (A=PU) polar decomposition is computed. Valid options: Right (A=UP), Left (A=PU). Default is 'right'.
        polar_ret_type (str, optional): The component of the decomposition to return. Valid options: Unitary/Orthonormal (U), Positive Semidefinite (P). Default is 'u'.

    Returns:
        list[list]: 2D array representing the requested component (U or P).
    """
    try:
        def to2d(x):
            return [[x]] if not isinstance(x, list) else x

        matrix = to2d(matrix)

        if not isinstance(matrix, list) or not matrix or not all(isinstance(row, list) for row in matrix):
            return "Error: matrix must be a non-empty 2D list"

        try:
            a = np.array(matrix, dtype=float)
        except (ValueError, TypeError):
            return "Error: matrix must contain numeric values"

        if not np.all(np.isfinite(a)):
            return "Error: matrix must contain only finite numbers"

        try:
            u_matrix, p_matrix = scipy_polar(a, side=polar_side.lower())
        except Exception as e:
            return f"Error: {str(e)}"

        if polar_ret_type.lower() == "u":
            return u_matrix.tolist()
        else:
            return p_matrix.tolist()

    except Exception as e:
        return f"Error: {str(e)}"

Online Calculator

matrix *

2D array of numeric values to decompose.

polar_side

Determines whether a right (A=UP) or left (A=PU) polar decomposition is computed.

polar_ret_type

The component of the decomposition to return.