Compute LU, QR, SVD, Cholesky & Eigendecompositions instantly with detailed step-by-step solutions. Understand the math, not just the answer.
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Choose a matrix decomposition method to learn more and use the dedicated calculator.
Factor a matrix into Lower and Upper triangular matrices. Essential for solving systems of linear equations efficiently.
Open calculator →Decompose into an orthogonal matrix Q and upper triangular R. Used in least squares regression and eigenvalue algorithms.
Open calculator →The most general matrix decomposition. Factorize any matrix into U, Σ, Vᵀ. Powers recommendation systems and data compression.
Open calculator →Efficient factorization for symmetric positive definite matrices into LLᵀ. Widely used in Monte Carlo simulations and optimization.
Open calculator →Find eigenvalues and eigenvectors. Fundamental to PCA, quantum mechanics, vibration analysis, and stability theory.
Open calculator →Reduce any square matrix to upper triangular form via orthogonal similarity. Reveals eigenvalues with numerical stability.
Open calculator →Factor symmetric matrices into LDLᵀ without square roots. Works for indefinite matrices where Cholesky fails.
Open calculator →Canonical form for any square matrix, including defective ones. Essential for solving systems of ODEs.
Open calculator →Every calculation shows the full derivation, not just the answer. Learn the algorithm as you go.
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Every decomposition is automatically verified by reconstructing the original matrix.
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Understanding matrix factorization is often the hardest part of linear algebra. While most calculators just give you the answer, Decomposition.ai acts as your personal AI math tutor. We don't just show the matrices; we explain the "why" behind every row swap and multiplier.
Our algorithms are designed to mimic human learning. See exactly how Gaussian elimination builds an LU decomposition or how Gram-Schmidt orthogonalizes a basis.
Confused by a result? Use our integrated AI chat to ask questions like "Why is this matrix singular?" or "What does this eigenvalue tell me about stability?"
Matrix decomposition (also called matrix factorization) is the process of breaking a matrix into a product of simpler matrices. Just as integers can be factored into primes, matrices can be factored into structured components that reveal their fundamental properties.
Decompositions are the backbone of numerical linear algebra. They power everything from solving systems of equations to machine learning algorithms, image compression, and quantum physics simulations.
The best decomposition depends on your matrix and your goal: