Search results
Results from the WOW.Com Content Network
That is, denoting each complex number by the real matrix of the linear transformation on the Argand diagram (viewed as the real vector space ), affected by complex -multiplication on . Thus, an m × n {\displaystyle m\times n} matrix of complex numbers could be well represented by a 2 m × 2 n {\displaystyle 2m\times 2n} matrix of real numbers.
under which addition and multiplication of complex numbers and matrices correspond to each other. For example, 2-by-2 rotation matrices represent the multiplication with some complex number of absolute value 1, as above. A similar interpretation is possible for quaternions [77] and Clifford algebras in general.
Specifically, the singular value decomposition of an complex matrix is a factorization of the form =, where is an complex unitary matrix, is an rectangular diagonal matrix with non-negative real numbers on the diagonal, is an complex unitary matrix, and is the conjugate transpose of . Such decomposition ...
A complex number can be visually represented as a pair of numbers (a, b) forming a vector on a diagram called an Argand diagram, representing the complex plane. Re is the real axis, Im is the imaginary axis, and i is the "imaginary unit", that satisfies i 2 = −1.
In mathematics, a Hermitian matrix (or self-adjoint matrix) is a complex square matrix that is equal to its own conjugate transpose—that is, the element in the i-th row and j-th column is equal to the complex conjugate of the element in the j-th row and i-th column, for all indices i and j: = ¯
Over the real or complex numbers, continuity implies that adj(A) commutes with B even when A is not invertible. Finally, there is a more general proof than the second proof, which only requires that an n × n matrix has entries over a field with at least 2n + 1 elements (e.g. a 5 × 5 matrix over the integers modulo 11).
The split-complex number = + can be represented by the matrix (). Addition and multiplication of split-complex numbers are then given by matrix addition and multiplication. The squared modulus of z is given by the determinant of the corresponding matrix.
where denotes the transpose of and is a fixed nonsingular, skew-symmetric matrix.This definition can be extended to matrices with entries in other fields, such as the complex numbers, finite fields, p-adic numbers, and function fields.