Compute LU, QR, SVD, and Cholesky decompositions with step-by-step explanations and factor matrix visualization.
Depending on its shape and properties, a matrix entered here can be analyzed with one or more of four factorizations: LU for triangular factors, QR for orthogonal and triangular factors, SVD for singular vectors and singular values, or Cholesky for a triangular factor paired with its transpose. The tool shows the factor matrices, explanatory steps, and a visual representation of their entries. The main source of confusion is that these methods are not interchangeable: Cholesky has stricter input requirements, while pivoting, signs, ordering, and reduced versus full forms can make other valid results look different.
A concrete input and expected output from the current implementation.
Input
Decomposition: Cholesky Matrix: 4 0 0 9
Expected output
L = 2 0 0 3 L^T = 2 0 0 3 L L^T = 4 0 0 9
The positive diagonal entries have square roots 2 and 3. Multiplying the lower-triangular factor by its transpose gives diagonal entries 2 squared = 4 and 3 squared = 9, with zero off-diagonal entries.
Supply the matrix as rectangular rows of numeric entries, then select LU, QR, SVD, or Cholesky. The selected method determines the factor identity to display: an LU result uses triangular factors, QR separates an orthogonal factor from an upper-triangular factor, SVD separates two singular-vector factors and a diagonal singular-value factor, and Cholesky produces a triangular factor paired with its transpose.
The step display identifies the factors and the matrix equation they satisfy. Read that equation carefully because LU may include a permutation matrix, and QR or SVD may be shown in reduced or full form. Factor visualization helps compare shapes and entry magnitudes, but correctness is determined by multiplying the factors in the stated order and reconstructing the original matrix within the displayed numerical precision.
Different answers can represent the same decomposition. LU may expose row permutations differently, QR columns can have opposite signs when the matching rows of R also change sign, and SVD singular-vector pairs can change signs together. SVD implementations may also order singular values differently. Compare the stated factor equation and reconstruct the input instead of comparing every factor entry directly.
No. SVD is defined for any real rectangular matrix, and QR also supports rectangular matrices, although the reported factor shapes can depend on whether a full or reduced form is used. Cholesky requires a real symmetric positive-definite matrix. LU may need row permutations, so include any displayed permutation factor when checking whether its product reconstructs the original matrix.
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