add QR Decomposition of matrix in linear algebra

TheAlgorithms · BhaktiVagadia · May 7, 2026 · May 7, 2026 · May 7, 2026 · May 7, 2026
commit 4047a43539be76ad838f937051da5f701fcac0c4
diff --git a/linear_algebra/qr_decomposition.py b/linear_algebra/qr_decomposition.py
@@ -0,0 +1,92 @@
+"""
+In linear algebra, a QR decomposition, also known as a QR factorization or QU factorization,
+is a decomposition of a matrix A into a product A = QR
+of an orthonormal matrix Q and an upper triangular matrix R.
+QR decomposition is often used to solve the linear least squares (LLS) problem
+and is the basis for a particular eigenvalue algorithm, the QR algorithm.
+
+This algorithm will simply attempt to perform QR decomposition on any square matrix.
+
+Reference: https://en.wikipedia.org/wiki/QR_decomposition
+"""
+
+from __future__ import annotations
+import numpy as np
+from scipy.linalg import qr
+
+def qr_decomposition(A: np.ndarray) -> tuple[np.ndarray, np.ndarray]:
+    """
+    Perform QR decomposition on a given matrix and raises an error if in
+    m×n matrix A if m is smaller than n or m,n is less than 2
+
+    >>> A = np.array([[1, 2, 3], [4, 5, 9], [7, 8, 15]])
+    >>> Q,R = qr_decomposition(A)
+    >>> Q
+    array([[-0.17,  0.9 ,  0.41],
+           [-0.51,  0.28, -0.82],
+           [-0.85, -0.35,  0.41]])
+    >>> R
+    array([[-17.75,  -9.63,  -8.11],
+           [  0.  ,   0.41,  -0.41],
+           [  0.  ,   0.  ,   0.  ]])
+    >>> A = np.array([[1, 2], [4, 5], [7, 8]])
+    >>> Q,R = qr_decomposition(A)
+    >>> Q
+    array([[-0.21,  0.89,  0.41],
+           [-0.52,  0.25, -0.82],
+           [-0.83, -0.38,  0.41]])
+    >>> R
+    array([[-9.64, -8.09],
+           [ 0.  , -0.76],
+           [ 0.  ,  0.  ]])
+    >>> A = np.array([[1, 2, 3], [4, 5, 6]])
+    >>> Q,R = qr_decomposition(A)
+    Traceback (most recent call last):
+        ...
+    ValueError: row size should be greater than column size
+    >>> A = np.array([[1], [4]])
+    >>> Q,R = qr_decomposition(A)
+    Traceback (most recent call last):
+        ...
+    ValueError: row size and column size should be greater than 2
+    >>> A = np.array([[1,4]])
+    >>> Q,R = qr_decomposition(A)
+    Traceback (most recent call last):
+        ...
+    ValueError: row size should be greater than column size
+    """
+
+
+    rows, columns = np.shape(A)
+    if rows < columns:
+        msg = (
+            "row size should be greater than column size"
+        )
+        raise ValueError(msg)
+    if rows < 2 or columns < 2:
+        msg = (
+            "row size and column size should be greater than 2"
+        )
+        raise ValueError(msg)
+    # Perform QR decomposition with pivoting
+    # Q: Orthogonal matrix
+    # R: Upper triangular matrix
+    # P: Pivot indices (permutation vector)
+
+    Q, R, P = qr(A, pivoting=True)
+
+    # Note: The bottom row of R is all zeros because the matrix is rank-deficient.
+    # Verification: A[:, P] should equal Q @ R
+    AP = A[:, P]
+    if(np.allclose(AP, Q @ R)):
+        return np.round(Q,2), np.round(R,2)
+    else:
+        msg = (
+            "No matrix found which decompose given matrix"
+        )
+        raise ValueError(msg)
+
+if __name__ == "__main__":
+    import doctest
+
+    doctest.testmod()