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Any real or complex matrix can be decomposed into a product of three matrices: a unitary matrix, a diagonal matrix of singular values, and the conjugate transpose of a unitary matrix. It reveals the intrinsic geometric structure of the data the matrix represents.

This is another name for the rational canonical form. It is a type of normal form where a matrix is similar to a block diagonal matrix, with companion matrices formed from the coefficients of the invariant polynomials.

A matrix decomposition where a matrix is written as the product of its eigenvectors and eigenvalues. If matrix $A$ is diagonalizable, then $A = PDP^{-1}$ where $P$ contains the eigenvectors and $D$ is the diagonal matrix of eigenvalues.

A matrix is unitarily similar to an upper triangular matrix called the Schur form. The eigenvalues of the original matrix appear on the diagonal of the Schur form.

A decomposition of a Hermitian, positive-definite matrix into the product of a lower triangular matrix and its conjugate transpose. Denoted by $A = LL^*$, where $L$ is the lower triangular matrix with real and positive diagonal entries.

Similar to the rational canonical form, it is used for matrices over the real numbers and consists of blocks that are either 1x1 or 2x2, representing real eigenvalues or pairs of complex conjugate eigenvalues, respectively.

A square matrix is similar to a block diagonal matrix where each block is a companion matrix of the invariant factors of the matrix. The rational canonical form shows the minimal polynomial structure of a linear transformation.

Every matrix can be factored into the product of an orthogonal matrix Q and an upper triangular matrix R. This decomposition is useful for solving linear systems and for eigenvalue computations.

A matrix is diagonalizable if it is similar to a diagonal matrix, meaning that it has a full set of linearly independent eigenvectors. Diagonal matrices have eigenvalues on the diagonal and zeroes elsewhere.

A symmetric matrix is positive definite if all its eigenvalues are positive, which is also equivalent to having all leading principal minors being positive and to the existence of a Cholesky decomposition.

Every square matrix is similar to a Jordan matrix, which is a block diagonal matrix where each block is a Jordan block. A Jordan block is an upper triangular matrix with equal eigenvalues on the diagonal and ones on the superdiagonal.

A Gram matrix is the result of multiplying a matrix by its transpose. It is symmetric, and when the original matrix has linearly independent columns, the Gram matrix is positive definite.

A square matrix can be expressed as the product of a lower triangular matrix L and an upper triangular matrix U. It facilitates the solution of systems of linear equations and the computation of determinants.

A Hermitian matrix is a complex square matrix that is equal to its own conjugate transpose. These matrices are always diagonalizable, and their eigenvalues are real.