Matrix Calculator: Determinant, Inverse, Eigenvalues & SVD
Determinant with cofactor expansion, inverse, symmetric eigen-decomposition and singular values.
Step-by-step guides
- How to Combine Transformation MatricesCombine transformation matrices by multiplying them in reverse application order for column vectors, including translations with homogeneous coordinates.
- What Conditions Make the Product of Two Matrices Zero?For compatible matrices, AB = 0 exactly when the image of B lies in the kernel of A; this also implies a rank bound.
- How to Calculate a Determinant by First-Row Cofactor ExpansionExpand along the first row by multiplying each entry by its signed minor. See the alternating signs, a worked 3×3 example, and common errors.
- How to Write Least Squares in Matrix FormExpress least squares as minimizing the residual norm ||Ax − b||, derive the normal equations, and see a fully worked linear-fit example.
- What Is the Relationship Between Eigendecomposition and SVD?The SVD is linked to the eigendecompositions of A* A and A A*: their eigenvectors give singular vectors and their eigenvalues are squared singular values.
- What Is the Intuitive Relationship Between SVD and PCA?For a centered data matrix, PCA directions are the right singular vectors of its SVD; squared singular values give component variances.
- Inverting a Gaussian (RBF) Kernel Matrix Without Numerical TroubleA Gaussian kernel matrix is positive definite for distinct points, so it is invertible, but it is often badly conditioned. Use a ridge term and a Cholesky solve instead.