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QR decomposition calculator

Factor a matrix A into Q·R, where Q has orthonormal columns and R is upper triangular, using the modified Gram-Schmidt process. Enter A on an editable grid (rows ≥ columns), and it returns Q, R and the product Q·R so you can check the factorization at a glance. QR is the workhorse behind least-squares regression and eigenvalue algorithms.

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QR decomposition calculator works straight from this page — free, instant, nothing to install. It sits under Matrices & vectors in our catalogue, alongside Cholesky decomposition calculator and LU Decomposition Calculator.

How to use it

  1. Open the tool — no signup or install needed.
  2. Enter your input or adjust the available options.
  3. Get your result instantly, then copy or download it.

Frequently asked questions

What is QR decomposition calculator?

Factor a matrix A into Q·R, where Q has orthonormal columns and R is upper triangular, using the modified Gram-Schmidt process. Enter A on an editable grid (rows ≥ columns), and it returns Q, R and the product Q·R so you can check the factorization at a glance. QR is the workhorse behind least-squares regression and eigenvalue algorithms.

What does a concrete case look like?

[[0, 1], [1, 0]] → Q = [[0, 1], [1, 0]], R = I — the tool shows every step in between, not just the final figure.

When would I actually use this?

Checking a decomposition done by hand, solving a linear system, and finding the eigenvalues that describe how a transformation stretches space.

What is the most common mistake?

Inverting a matrix to solve a system. Elimination or a factorisation is both faster and far more stable numerically; an explicit inverse amplifies rounding error, especially when the matrix is close to singular.

How is QR decomposition calculator different from Cholesky decomposition calculator?

They sit next to each other but answer different questions: Cholesky decomposition calculator is the one to open when you need it to factor a symmetric positive-definite matrix A into L·Lᵀ, where L is lower triangular — the Cholesky decomposition. Enter A on an editable square grid and it returns L, Lᵀ and the product L·Lᵀ; if the matrix is not symmetric or not positive-definite, it tells you which condition failed. Twice as fast as LU, Cholesky is used in optimization, Kalman filters and Monte-Carlo simulation. Pick whichever matches what you're starting from — both are free.

Is there a tool for the next step?

LU Decomposition Calculator is the closest one after this: Factor a square matrix into a lower L and upper U triangular matrix (Doolittle method).

What else is worth having open alongside it?

SVD calculator (singular value decomposition) and Partial Fraction Decomposition Calculator — they come up in the same task often enough to be worth a second tab.

Where do the figures come from?

The decompositions follow their textbook definitions with partial pivoting where it applies. Results are exact for small integer matrices and subject to floating-point rounding otherwise.

Further reading

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