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
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- Basic terminology and notationBasic terminology and notation for matrices establishes the foundational language used throughout linear algebra. A...
- Operations on matricesOperations on matrices are the fundamental ways of combining and transforming matrices. The main operations include:...
- Particular types of matricesParticular types of matrices are special matrices with specific structural properties or algebraic characteristics...
- Matrix equivalenceMatrix equivalence is a relation between matrices of the same size that captures when two matrices represent the same...
- Canonical forms of matricesCanonical forms of matrices are standard representatives for equivalence or similarity classes of matrices. A...
- Factorization of matricesMatrix factorization (or matrix decomposition ) expresses a matrix as a product of simpler matrices, revealing...
- Similarity of matricesSimilarity is a relation between square matrices that captures when two matrices represent the same linear operator...
- Nonsingular matrices and equivalencesNonsingular matrices (also called invertible or nondegenerate matrices) are square matrices that have a...
- Rank and nullityRank and nullity measure the dimensions of key subspaces associated with a matrix. The rank is the dimension of the...
- Eigenvalues and eigenvectorsEigenvalues and eigenvectors capture the intrinsic stretching/compressing directions of a linear transformation. For...
- DeterminantsDeterminants are scalar values computed from square matrices that encode important properties. The determinant...
- Choosing factorizations and understanding numerical errorExact algebra asks whether a solution exists. Numerical linear algebra also asks how errors in data and arithmetic...
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