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In mathematics, a triangular matrix is a special kind of square matrix. A square matrix is called lower triangular if all the entries above the main diagonal are zero.
The Jordan normal form is the most convenient for computation of the matrix functions (though it may be not the best choice for computer computations). Let f(z) be an analytical function of a complex argument. Applying the function on a n×n Jordan block J with eigenvalue λ results in an upper triangular matrix:
Every complex-valued square matrix , regardless of diagonalizability, has a Schur decomposition given by = where is upper triangular and is unitary (meaning =). The eigenvalues of A {\displaystyle A} are exactly the diagonal entries of U {\displaystyle U} ; if at most one of them is zero, then the following is a square root [ 7 ]
Triangular matrix: A matrix with all entries above the main diagonal equal to zero (lower triangular) or with all entries below the main diagonal equal to zero (upper triangular). Tridiagonal matrix: A matrix with the only nonzero entries on the main diagonal and the diagonals just above and below the main one. X–Y–Z matrix
In linear algebra, the Cholesky decomposition or Cholesky factorization (pronounced / ʃ ə ˈ l ɛ s k i / shə-LES-kee) is a decomposition of a Hermitian, positive-definite matrix into the product of a lower triangular matrix and its conjugate transpose, which is useful for efficient numerical solutions, e.g., Monte Carlo simulations.
One can always write = where V is a real orthogonal matrix, is the transpose of V, and S is a block upper triangular matrix called the real Schur form. The blocks on the diagonal of S are of size 1×1 (in which case they represent real eigenvalues) or 2×2 (in which case they are derived from complex conjugate eigenvalue pairs).
A quasi-triangular matrix is a matrix that when expressed as a block matrix of 2 × 2 and 1 × 1 blocks is triangular. This is a stronger property than being Hessenberg . Just as in the complex case, a family of commuting real matrices { A i } may be simultaneously brought to quasi-triangular form by an orthogonal matrix.
A packed storage matrix, also known as packed matrix, is a term used in programming for representing an matrix. It is a more compact way than an m-by-n rectangular array by exploiting a special structure of the matrix. Typical examples of matrices that can take advantage of packed storage include: