About QR Decomposition Calculator
This calculator factors a matrix with independent columns as A = QR, where Q has orthonormal columns and R is upper triangular. For [[3, 1], [4, 2]] it finds Q = [[3/5, −4/5], [4/5, 3/5]] and R = [[5, 11/5], [0, 2/5]]. It is free, needs no sign-up and runs in your browser, so nothing you enter is uploaded.
It uses the Gram-Schmidt process: take each column of A, subtract its projection onto the directions already found, then divide by the length. The working shows each remaining direction as a vector together with its squared length, so you can follow the arithmetic before the square roots appear.
Entries are kept exact, so a result such as 1 ÷ √2 is written as a square root rather than rounded. The matrix may be taller than it is wide, up to 6 × 6, but its columns must be linearly independent; otherwise the calculator says so instead of dividing by zero.
How to use QR Decomposition Calculator
- Choose an operation, such as Determinant, Inverse or A × B.
- Set the size of matrix A with the row and column selectors. Square operations use a square grid, and two-matrix operations also show matrix B.
- Type the entries. Use integers, decimals or fractions like 3/4, and leave a cell blank for zero.
- The tool opens on a worked example. Read the result, typeset as a determinant, a rank or a matrix.
- Read the steps for the working, and use the decimal switch if you would rather see decimals than fractions.
Features
- Grids up to 6 × 6, with rows and columns set separately for non-square matrices.
- Addition, subtraction and multiplication of two matrices, with a clear message when the sizes do not match.
- Determinant with the 2 × 2 rule, first-row expansion for 3 × 3 and row reduction for larger matrices.
- Inverse by row-reducing [A | I], with a plain message when the matrix is singular.
- Rank and reduced row echelon form with each row operation listed.
- Adjugate and cofactor matrix for square matrices.
- Eigenvalues from the characteristic polynomial, including complex ones, with eigenvectors: exact for rational eigenvalues and numeric (labelled) for irrational ones.
- LU decomposition (with row swaps when a pivot is zero) and QR decomposition by Gram-Schmidt, exact with square roots.
- Fraction entries and fraction results, with a switch to decimals that also applies to the steps.
- Transpose of any matrix.
Gram-Schmidt in short
Each new direction is the column minus its projections onto the earlier directions.
u_k = a_k − Σ (a_k · u_j / u_j · u_j) u_j
q_k = u_k / ‖u_k‖
R[i][j] = q_i · a_j (j ≥ i)Worked examples
[[3, 1], [4, 2]] gives Q = [[3/5, −4/5], [4/5, 3/5]], R = [[5, 11/5], [0, 2/5]]
The first column (3, 4) has length 5. Removing its direction from (1, 2) leaves (−8/25, 6/25), of length 2/5.
[[1, 1], [1, −1]] gives Q = (1/√2)[[1, 1], [1, −1]], R = [[√2, 0], [0, √2]]
The two columns are already perpendicular, so only the lengths (both √2) are divided out.
Privacy: how it works
- All the arithmetic runs in your browser tab. Your matrices are not sent to Ulkaa or anyone else, and there is nothing to sign in to.
Tips
- Write fractions as 3/4 rather than 0.75 if you want results to stay exact.
- Leave entries blank for zero instead of typing every zero in a large grid.
- To solve a linear system, put the coefficients and constants in one grid and use RREF, or use the equation solver.
- Multiplication is not commutative. To check AB against BA, swap which matrix is A and which is B.
Frequently asked questions
Is it really free?
Yes. There is no price, no trial and no credit card. Ulkaa Tool does the work with your own device, so there is no server bill to pass on to you.
Do I need an account?
No. There is no sign-up and no login. Open the page and use it. An Ulkaa account is not needed for anything on this tool.
Does it run in my browser?
Yes. Your matrices are processed by code running in your browser tab, on your own device.
Is my data uploaded?
No. Your matrices are never sent to Ulkaa or to anyone else. You can check this yourself by watching the Network tab in your browser developer tools while you use the tool.
What is QR decomposition?
It writes a matrix as a product of an orthogonal matrix Q (columns of length 1 at right angles) and an upper triangular matrix R. It is the basis of stable least-squares fitting and of the QR algorithm for eigenvalues.
Why does it say the columns are dependent?
Gram-Schmidt needs each column to add a new direction. If one column is a combination of the others its remainder is zero, there is nothing to normalise, and this reduced QR form does not exist.
Are the numbers exact?
Yes. Fractions stay exact and square roots are kept as square roots. Use the decimal switch to see approximate decimal entries instead.
More Matrix Calculator pages
- Matrix CalculatorAdd, multiply, invert, decompose (LU, QR) and analyse matrices up to 6 × 6, with typeset working.
- Matrix Determinant CalculatorFind the determinant of any square matrix up to 6 × 6.
- Inverse Matrix CalculatorFind A⁻¹ for a square matrix, or learn why it does not exist.
- Matrix Multiplication CalculatorMultiply matrix A by matrix B.
- Matrix Addition CalculatorAdd or subtract two matrices of the same size.
- Matrix Transpose CalculatorSwap the rows and columns of a matrix.
- Matrix Rank CalculatorFind how many independent rows a matrix has.
- Eigenvalue CalculatorFind the eigenvalues of a square matrix and see the characteristic polynomial.
- 2x2 Matrix Determinant CalculatorEnter a, b, c and d to get ad − bc.
- 3x3 Matrix Determinant CalculatorExpand a 3 × 3 determinant along the first row with every minor shown.
- 3x3 Inverse Matrix CalculatorInvert a 3 × 3 matrix with every row operation listed.
- RREF CalculatorRow-reduce any matrix up to 6 × 6 to reduced row echelon form.
- Adjugate and Cofactor Matrix CalculatorGet the cofactor matrix, its transpose the adjugate, and the determinant.