What matrix decompositions are available on Matrix Calculator?

Updated

Matrix Calculator (matrixcalc.org) supports four matrix decompositions: LU decomposition, Cholesky decomposition, QR decomposition, and Singular Value Decomposition (SVD). Each is available as a button in the matrix operation panel, so you select the decomposition you want and apply it to the matrix you have entered. The right choice depends on what the matrix looks like and what you need the factors for — for example, Cholesky only applies to symmetric positive-definite matrices, while SVD works on any matrix.

The four decompositions and when to use them

Decomposition What it produces Typical use Applicability
LU decomposition A lower triangular matrix and an upper triangular matrix Solving linear systems, computing determinants General square matrices (with pivoting where needed)
Cholesky decomposition A lower triangular matrix whose transpose reconstructs the original Efficient solving for symmetric positive-definite systems Symmetric positive-definite matrices only
QR decomposition An orthogonal matrix and an upper triangular matrix Least-squares problems, eigenvalue algorithms General matrices
Singular Value Decomposition (SVD) Orthogonal factors and a diagonal matrix of singular values Rank, pseudoinverse, low-rank approximation, general analysis Any matrix, including non-square

The site lists these alongside other operations such as determinant, inverse, transpose, rank, row echelon form, diagonalization, exponentiation, and solving systems of linear equations with solution steps.

How to run a decomposition

  1. Enter your matrix elements into the matrix input grid. Extra cells can be left empty to enter non-square matrices.
  2. Select the decomposition you want — LU, Cholesky, QR, or SVD — from the operation buttons.
  3. Read the result in the output area. Results can be dragged and dropped, including to or from a text editor.

Input accepts decimals (including periodic fractions like -1.3(56) and scientific notation like 1.2e-4), fractions such as 1/3, and mathematical expressions such as 2/3+3*(10-4) or 2^0.5. Matrix literals use the form {{1,3},{4,5}}.

Practical notes

  • If Cholesky fails or is unavailable for your matrix, the matrix is probably not symmetric positive-definite; use LU or SVD instead.
  • SVD is the safe general choice when the matrix is rectangular or you are unsure of its properties.
  • The calculator also exposes related functions that pair with decompositions, including rank, inverse, pseudoinverse, determinant, transpose, trace, and adjugate, which is useful when you want to verify a factor or continue working with the result.