Linear Algebra Reference
Reference for vectors, matrices, eigenvalues, and linear transformations.
Vectors and Matrices
Vectors are elements of a vector space with addition and scalar multiplication. Matrices represent linear transformations and systems of equations. Matrix multiplication is associative but not commutative. The determinant det(A) is zero iff A is singular (non-invertible). The inverse A^(-1) exists iff det(A) ≠ 0. The transpose A^T swaps rows and columns. The rank is the dimension of the column/row space. Trace is the sum of diagonal elements. Matrix operations are fundamental to graphics, ML, and cryptography.
- determinant
- Zero = singular
- inverse
- A^(-1) exists
- rank
- Column space dim
invertibilitydet ≠ 0linear independenceEigenvalues and Decompositions
Eigenvalues λ and eigenvectors v satisfy Av = λv. The characteristic polynomial det(A - λI) = 0 finds eigenvalues. Eigendecomposition A = QΛQ^(-1) for diagonalizable A. SVD (Singular Value Decomposition) A = UΣV^T works for any matrix. QR decomposition factors into orthogonal Q and upper triangular R. LU decomposition factors into lower and upper triangular. Cholesky decomposition applies to positive definite matrices. These decompositions solve systems, compute determinants, and enable PCA. Numerical stability varies; SVD is most stable but expensive.
- eigenvalues
- Av = λv
- SVD
- A = UΣV^T
- QR
- Orthogonal * triangular
characteristic polyany matrixstable solve