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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:
The inverse of an upper triangular matrix, if it exists, is upper triangular. The product of an upper triangular matrix and a scalar is upper triangular. Together these facts mean that the upper triangular matrices form a subalgebra of the associative algebra of square matrices for a given size.
of a permutation matrix (which has exactly a single in each column, and otherwise zeros), a lower triangular matrix and an upper triangular matrix . The determinants of the two triangular matrices L {\displaystyle L} and U {\displaystyle U} can be quickly calculated, since they are the products of the respective diagonal entries.
For example, if A is a 3-by-0 matrix and B is a 0-by-3 matrix, then AB is the 3-by-3 zero matrix corresponding to the null map from a 3-dimensional space V to itself, while BA is a 0-by-0 matrix. There is no common notation for empty matrices, but most computer algebra systems allow creating and computing with them.
In numerical analysis and linear algebra, lower–upper (LU) decomposition or factorization factors a matrix as the product of a lower triangular matrix and an upper triangular matrix (see matrix multiplication and matrix decomposition). The product sometimes includes a permutation matrix as well.
Using row operations to convert a matrix into reduced row echelon form is sometimes called Gauss–Jordan elimination. In this case, the term Gaussian elimination refers to the process until it has reached its upper triangular, or (unreduced) row echelon form. For computational reasons, when solving systems of linear equations, it is sometimes ...
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).
Thus, the row echelon form can be viewed as a generalization of upper triangular form for rectangular matrices. A matrix is in reduced row echelon form if it is in row echelon form, with the additional property that the first nonzero entry of each row is equal to and is the only nonzero entry of its column. The reduced row echelon form of a ...