Linear algebra

Definitions

• Basis
• Column Space
• Determinant
• Eigenvalues and eigenvectors
• Factorizations
• Krylov Subspace
• Linear map
• Linear Systems
• Lᵖ Norm of a vector
• Matrix
• Matrix Condition Number
• Matrix Function
• Matrix similarity
• Matrix trace
• Matrix x Matrix multiplication
• Matrix x Vector multiplication
• Norm of an operator
• Rank
• Rayleigh quotient

Special Matrices

• Circulant matrix
• Covariance Matrix
• Fourier Matrix
• Haar Wavelet Matrix
• Heisenberg Matrix
• Hermitian Matrix
• Hessenberg Matrix
• KKT System Matrix
• Moore–Penrose Inverse
• Orthonormal matrix
• Permutation Matrix
• Positive Definite Matrix
• Positive Semi-Definite Matrix
• Symmetric Matrix
• Toeplitz Matrix
• Unitary Matrix
• Vandermonde Matrix

Norms

• Frobenius Norm
• Nuclear Norm
• Spectral Norm

Factorizations

• Cholesky Decomposition
• Column-Row Factorization (CR)
• Eigendecomposition
• Lower–Upper Factorization (LU)
• Polar Decomposition
• QR factorization
• Singular Value Decomposition (SVD)

Theorems

• Basis Theorem
• Circulant Matrix Representation as a Polynomial of Shifts
• Continuity and Boundedness of Linear Maps
• Convexity of -log(det(X))
• Eckart-Young Theorem
• Fundamental Theorem of Linear Algebra
• Spectral Theorem

Applications

• Big matrix sampling

Observations

• Free Parameters in an n x m Matrix of Rank r
• Small Matrix Perturbation
• Matrix multiplication order matters
• Why So Many Matrices Are of Low Rank