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Matrix Decomposition Calculator

Use this free matrix decomposition calculator to decompose a 2x2 matrix using LU, QR, or eigendecomposition. Select a method, enter four matrix entries, and get the complete factorization with step-by-step details.

Enter Values

Choose the factorization type based on your application

Top-left entry of the matrix

Top-right entry of the matrix

Bottom-left entry of the matrix

Bottom-right entry of the matrix

Result

Enter values above and click Calculate to see your result.

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Formula

A = LU (lower-upper), A = QR (orthogonal-upper), or A = PDP inverse (eigen)

LU decomposes into lower and upper triangular matrices. QR decomposes into an orthogonal matrix and upper triangular matrix. Eigendecomposition uses eigenvalues and eigenvectors to express A as PDP inverse.

Worked Example

A = [[4, 1], [2, 3]], Method: LU Step 1: L21 = a21/a11 = 2/4 = 0.5 Step 2: U11 = 4, U12 = 1 Step 3: U22 = a22 - L21 x U12 = 3 - 0.5(1) = 2.5 Step 4: L = [[1, 0], [0.5, 1]], U = [[4, 1], [0, 2.5]] Result: A = LU verified.

What Is Matrix Decomposition?

Matrix decomposition (factorization) breaks a matrix into a product of simpler matrices with special properties. Different decompositions serve different purposes. This is one of the most important techniques in computational linear algebra, used in everything from solving systems of equations to machine learning algorithms.
  • LU decomposition is used to solve systems of linear equations efficiently, especially when solving multiple systems with the same coefficient matrix
  • QR decomposition is the foundation of the QR algorithm for computing eigenvalues of large matrices
  • Eigendecomposition (PDP inverse) simplifies matrix powers, exponentials, and differential equations
  • LU requires a nonzero pivot (a11 not equal to 0). Partial pivoting can handle zero pivots
  • QR always exists for any matrix with linearly independent columns
  • Eigendecomposition requires the matrix to be diagonalizable (enough independent eigenvectors)

Each decomposition method has strengths and limitations. LU is fastest for solving linear systems. QR is more numerically stable. Eigendecomposition provides the deepest insight into the matrix structure but requires diagonalizability.

You can also calculate changes using our Eigenvalue Calculator, Diagonalization Calculator, 2x2 Determinant Calculator or Inverse Matrix Calculator.

Frequently Asked Questions

When should I use LU vs QR vs eigendecomposition?

Use LU for solving systems of equations (Ax = b) efficiently. Use QR for least-squares problems and when numerical stability matters. Use eigendecomposition when you need to understand the matrix structure, compute powers, or solve differential equations.

Why might LU decomposition fail?

Basic LU fails when a pivot element is zero (a11 = 0 for 2x2). In practice, LU with partial pivoting (row swapping) handles this by reordering rows to avoid zero pivots. This calculator reports when the basic LU cannot proceed.

What makes QR decomposition special?

Q is an orthogonal matrix (Q transpose = Q inverse), meaning it preserves lengths and angles. This makes QR numerically very stable. The QR algorithm, which repeatedly applies QR decomposition, is the standard method for computing eigenvalues of large matrices.

Can I decompose a singular matrix?

LU can decompose singular matrices (the result will have a zero on the diagonal of U). QR also works for singular matrices. Eigendecomposition may fail for singular matrices if they are not diagonalizable, though singular diagonalizable matrices do decompose.

What is the computational cost of each method?

For a 2x2 matrix, all methods are instant. For general n x n matrices: LU is O(n cubed / 3), QR via Gram-Schmidt is O(2 n cubed / 3), and full eigendecomposition is O(n cubed) with iterative refinement.

Related Calculators

Eigenvalue CalculatorDiagonalization Calculator2x2 Determinant CalculatorInverse Matrix Calculator

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