Qr Decomposition Calculator

How to use this QR decomposition calculator

What this calculator does

This calculator takes a real square matrix A and computes a QR factorization A = QR. The columns of Q are orthonormal, so QᵀQ = I, and R is upper triangular. The tool also reconstructs A from the factors and reports numerical errors. It is designed for small educational matrices and verification work; it is not a substitute for a production numerical linear algebra library when matrices are large, complex-valued, sparse, or extremely ill-conditioned.

When to use it

• Check a hand-worked QR decomposition from a linear algebra course.
• Prepare a small matrix for least-squares or orthogonal-basis calculations.
• Compare how changing one matrix entry affects Q, R, and numerical residuals.
• Export a reproducible workbook containing the inputs, factors, and verification table.

How to calculate

1. The calculator opens with a complete 3 × 3 demonstration matrix and an immediately available Excel workbook. Review the displayed Q and R matrices to see a valid result before editing.
2. Choose Matrix size as 2 × 2, 3 × 3, or 4 × 4. Changing the size preserves overlapping entries and creates or removes the remaining matrix cells.
3. Replace each Matrix A entry labeled aᵢⱼ. Results update live. Use a period for decimals; values such as -2.5 and 1e-3 are accepted.
4. Read the factorization status, Q, R, rank, residual, orthogonality error, and the reconstruction table. Then select Download Excel to create a validated OOXML workbook from the current canonical values.
5. Reset clears the demonstration and all matrix entries. Download Excel is then disabled until every required matrix cell contains a valid number and the matrix has linearly independent columns.

Input guide

Matrix size is required and determines the number of rows and columns. It accepts 2, 3, or 4. A 3 × 3 matrix is a realistic default for classroom work. Increasing the size introduces more columns that must remain linearly independent. A common mistake is to change the size and leave a new cell blank.

Matrix A entries are required finite real numbers. They are unitless unless your application assigns units consistently. Each field accepts ordinary decimal or scientific notation, for example 12, -51, or 4.2e2. Commas, currency symbols, percent signs, and nonnumeric text are rejected rather than silently altered. Changing any entry can modify every later column of Q and much of R. Duplicate or proportional columns make the matrix rank-deficient, so a full QR factorization with a positive diagonal in R cannot be produced.

Output guide

Factorization status states whether the current matrix has a valid full-rank QR decomposition. Orthonormal matrix Q contains normalized mutually perpendicular columns. Upper-triangular matrix R contains the projection coefficients and column norms; entries below its diagonal are zero by construction. The sign convention used here makes the diagonal of R nonnegative, which removes the usual sign ambiguity for full-rank matrices.

Rank is the number of independent columns detected by the factorization. Maximum |A – QR| is the largest absolute reconstruction difference across all entries. Maximum |QᵀQ – I| measures the largest departure from orthonormality. Values close to zero are desirable; tiny nonzero values are expected because JavaScript uses floating-point arithmetic. Minimum |diag(R)| is the smallest absolute diagonal entry in R and helps reveal a nearly dependent column. The Reconstruction detail table lists the original value, reconstructed value, signed difference, and absolute difference for every matrix position. These checks are numerical estimates, while the identity A = QR is exact in ideal arithmetic.

Worked example

The startup matrix is A = [12, – 51, 4], [6, 167, – 68], [ – 4, 24, – 41][6/7, 3/7, – 2/7] and r₁₁ = 14. Orthogonalizing the next two columns produces R = [14, 21, – 14], [0, 175, – 70], [0, 0, 35]