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The Schur complement arises when performing a block Gaussian elimination on the matrix M.In order to eliminate the elements below the block diagonal, one multiplies the matrix M by a block lower triangular matrix on the right as follows: = [] [] [] = [], where I p denotes a p×p identity matrix.
For second-order problems, such as the Laplace equation or linear elasticity, the matrix of the system has condition number of the order 1/h 2, where h is the characteristic element size. The Schur complement, however, has condition number only of the order 1/h.
In the decimal numbering system, the radix complement is called the ten's complement and the diminished radix complement the nines' complement. In binary, the radix complement is called the two's complement and the diminished radix complement the ones' complement. The naming of complements in other bases is similar.
In the mathematical fields of linear algebra and functional analysis, the orthogonal complement of a subspace of a vector space equipped with a bilinear form is the set of all vectors in that are orthogonal to every vector in .
There is also a real Schur decomposition. If A is an n × n square matrix with real entries, then A can be expressed as [4] = where Q is an orthogonal matrix and H is either upper or lower quasi-triangular. A quasi-triangular matrix is a matrix that when expressed as a block matrix of 2 × 2 and 1 × 1 blocks is
Suppose a vector norm ‖ ‖ on and a vector norm ‖ ‖ on are given. Any matrix A induces a linear operator from to with respect to the standard basis, and one defines the corresponding induced norm or operator norm or subordinate norm on the space of all matrices as follows: ‖ ‖, = {‖ ‖: ‖ ‖ =} = {‖ ‖ ‖ ‖:} . where denotes the supremum.
Thus, an matrix of complex numbers could be well represented by a matrix of real numbers. The conjugate transpose, therefore, arises very naturally as the result of simply transposing such a matrix—when viewed back again as an n × m {\displaystyle n\times m} matrix made up of complex numbers.
In characteristic 2 the latter equality turns into = {, …,} (¯) what therefore provides an opportunity to polynomial-time calculate the Hamiltonian cycle polynomial of any unitary (i.e. such that = where is the identity n×n-matrix), because each minor of such a matrix coincides with its algebraic complement: = (+ /) where ...