Matrices

Matrices are rectangular arrays of numbers, symbols, or expressions arranged in rows and columns. They are fundamental objects in linear algebra used to represent linear transformations, systems of linear equations, and data structures.

Matrices support operations such as addition, multiplication, transposition, and inversion (for nonsingular matrices). Special types of matrices—such as diagonal, symmetric, orthogonal, and triangular matrices—have important properties that simplify computations.

Key topics covered in this section include:

• Basic terminology (size, notation, special matrices)
• Operations (addition, multiplication, transpose, inverse)
• Special types (symmetric, orthogonal, Hermitian, nilpotent, etc.)
• Equivalence and similarity relations
• Factorizations (LU, QR, SVD, Cholesky, etc.)
• Rank, nullity, and the rank-nullity theorem
• Eigenvalues and eigenvectors
• Determinants and trace