Symmetric matrix

A real square matrix is symmetric when it equals its transpose, so its entry in row i and column j matches the entry in row j and column i.

Matrix transpose: The transpose of a matrix is formed by exchanging its rows and columns. Symmetry is defined by equality with this operation.
Spectral theorem: A theorem stating that real symmetric matrices admit an orthonormal basis of eigenvectors. It gives the central structural decomposition of symmetric matrices.
Covariance matrix: A matrix whose entries are covariances between pairs of random variables. Covariance matrices are symmetric because covariance is unchanged when its arguments are exchanged.
Skew-symmetric matrix: A square matrix A satisfying Aᵀ = −A. It reverses the transpose relation that defines symmetry.
Square matrix: A matrix with the same number of rows and columns. Only square matrices can equal their own transpose.
Orthogonal diagonalization: A factorization of a matrix as QDQᵀ, with Q orthogonal and D diagonal. Every real symmetric matrix has this form.
Positive-definite matrix: A symmetric matrix whose quadratic form is positive for every nonzero vector. Positive definiteness is a key additional property in optimization and statistics.
Hermitian matrix: A complex square matrix equal to its conjugate transpose. It extends real symmetry to matrices with complex entries.
Matrix entry: A scalar occupying a specified row and column of a matrix. Symmetry pairs the entries at positions (i,j) and (j,i).
Orthogonal matrix: A square matrix whose transpose is its inverse. Its columns form the orthonormal eigenbasis used in diagonalization.

Linked from 9 pages