Qr Decomposition Calculator (2x2)

QR Decomposition for 2x2 Matrices

QR decomposition factors any matrix A into an orthogonal matrix Q and an upper triangular matrix R, expressed as A = QR. For a 2x2 matrix, this factorization is computed analytically using the Gram-Schmidt orthogonalization process, which transforms the column vectors of A into a mutually perpendicular, unit-length basis. Applications span numerical linear algebra, least-squares regression, eigenvalue computation, and signal processing, making QR decomposition one of the most practically important tools in applied mathematics.

The Core Formula and Properties

The decomposition satisfies three conditions simultaneously:

• A = QR: The original matrix equals the product of Q and R.
• Q^TQ = I: Q is orthogonal; its columns form an orthonormal set.
• R is upper triangular: The element R[2,1] = 0 by construction, leaving R[1,1] = r₁₁, R[1,2] = r₁₂, and R[2,2] = r₂₂ as the three non-zero entries.

Because Q preserves vector norms (||Qx|| = ||x|| for any vector x), QR-based solvers avoid the numerical condition-squaring problem that afflicts the normal equations in least-squares regression.

Numerical Stability and Practical Considerations

QR decomposition excels in numerical computations because the orthogonal matrix Q preserves vector norms and angles, protecting against condition-number squaring. This is critical in least-squares problems: computing A^TA directly squares the condition number κ(A) to κ(A)², dramatically amplifying rounding errors on ill-conditioned systems. QR decomposition instead maintains condition number κ(A) throughout the solution process. For small matrices like 2x2, the Gram-Schmidt algorithm is highly efficient and analytically transparent, making it ideal for educational purposes and analytical verification. The orthogonality properties ensure that solutions remain accurate even when the original matrix is poorly conditioned or nearly singular.

Gram-Schmidt Derivation: Step by Step

Given matrix A with first column a₁ = [a₁₁, a₂₁]^T and second column a₂ = [a₁₂, a₂₂]^T, the classical Gram-Schmidt algorithm proceeds as follows:

1. Compute r₁₁: r₁₁ = ||a₁|| = sqrt(a₁₁² + a₂₁²). This is the Euclidean norm of the first column and equals R[1,1].
2. Normalize to get q₁: q₁ = a₁ / r₁₁ = [a₁₁/r₁₁, a₂₁/r₁₁]^T. This becomes the first column of Q.
3. Compute r₁₂: r₁₂ = q₁ · a₂ = (a₁₁ × a₁₂ + a₂₁ × a₂₂) / r₁₁. This scalar projection equals R[1,2].
4. Orthogonalize the second column: a_2,⊥ = a₂ − r₁₂ × q₁. Subtracting the projection removes the component of a₂ parallel to q₁.
5. Compute r₂₂: r₂₂ = ||a_2,⊥||. This equals R[2,2] and is zero only when A is singular.
6. Normalize to get q₂: q₂ = a_2,⊥ / r₂₂. This becomes the second column of Q.

Variable Definitions

• a₁₁: Row 1, column 1 of A (top-left input)
• a₁₂: Row 1, column 2 of A (top-right input)
• a₂₁: Row 2, column 1 of A (bottom-left input)
• a₂₂: Row 2, column 2 of A (bottom-right input)
• Q: 2x2 orthogonal matrix with orthonormal columns q₁ and q₂
• R: 2x2 upper triangular matrix with diagonal entries r₁₁ and r₂₂

Worked Example

Let A = [[3, 1], [4, 1]].

• r₁₁ = sqrt(3² + 4²) = sqrt(25) = 5
• q₁ = [3/5, 4/5] = [0.6, 0.8]
• r₁₂ = (3 × 1 + 4 × 1) / 5 = 7/5 = 1.4
• a_2,⊥ = [1 − 1.4 × 0.6, 1 − 1.4 × 0.8] = [0.16, −0.12]
• r₂₂ = sqrt(0.16² + 0.12²) = sqrt(0.04) = 0.2
• q₂ = [0.16/0.2, −0.12/0.2] = [0.8, −0.6]

Result: Q = [[0.6, 0.8], [0.8, −0.6]], R = [[5, 1.4], [0, 0.2]]. Verification: Q × R = [[0.6×5 + 0.8×0, 0.6×1.4 + 0.8×0.2], [0.8×5 + (−0.6)×0, 0.8×1.4 + (−0.6)×0.2]] = [[3, 1], [4, 1]] = A. Confirmed.

Applications and Methodology Sources

• Least-squares regression: Solving min||Ax − b|| via back-substitution on Rx = Q^Tb is more stable than the normal equations (see Georgia Tech least-squares lecture notes).
• Eigenvalue algorithms: The iterative QR algorithm repeatedly factors A to converge on eigenvalues, as described in UC Berkeley Python Numerical Methods.
• Signal processing: Q matrices appear in orthogonal transforms that preserve signal energy and are essential in spectral analysis and filter design.
• Geometric computations: QR decomposition provides an efficient basis for computing projections, rotations, and reflections in computer graphics and geometric algorithms.

The Gram-Schmidt procedure implemented in this calculator follows the derivation in UCLA Math 151B — QR Decomposition with Gram-Schmidt and the factorization conventions established in UC Davis Linear Algebra course notes.