In linear algebra, the adjugate or classical adjoint of a square matrix isthe transpose of its cofactor matrix. ... The adjugate has sometimes been called the "adjoint", but today the "adjoint" of a matrix normally refers to its corresponding adjoint operator, which is its conjugate transpose...
In mathematics, a matrix is a powerful tool that has found useful applications in engineering, physics, economics, etc. Matries form algebraic systems in which many of the ordinary laws of arithmetic holds true. It is composed of numbers (real or conjugate) symbols, or expressions ordered in ...
What is the conjugate transpose of the following matrix? $$\displaystyle \begin{aligned} A &= \begin{pmatrix} 1 & i \\ 1 & i \end{pmatrix}. \end{aligned} $$ (2.7) The conjugate transpose of a matrix is found using the following two steps. First, we “conjugate” the comple...
and the matrix exponential of a diagonal matrix is simply the exponential function applied to the diagonal entries. But not all matrices are diagonalizable! The solution that is usually presented in the classroom is to use the Jordan canonical form instead and to compute the matrix exponential of ...
The matrix element of p is ##p_{ij} = -i\hbar \int u_j^*(x)\frac{\partial}{\partial x} u_i(x) \, dx##. The complex conjugate of this is ##\bar{p}_{ij} = i\hbar \int u_j(x)\frac{\partial}{\partial x} u_i^*(x) \, dx \neq -p_{ij}##. What does an ...
Cartesian coordinates is n_{Cart} = U^{-1} n_{diag} U. (For my matrix, the inverse performs the same operation that the complex conjugate transpose does.) When I plot each of these elements (as a function of their position in the cross-section of the structure) I find the f...
For instance, if the canonical line bundle is ample (or more generally, nef), this volume is equal to the intersection number (roughly speaking, the number of common zeroes of generic sections of the canonical line bundle); this is a special case of the asymptotic Riemann-Roch theorem. In...
Note that the integrand here is not complex analytic due to the presence of the complex conjugates. However, this can be dealt with by the trick of replacing the complex conjugate by a variable which is formally conjugate to , but which is allowed to vary independently of . More precisely,...
(linear algebra) (Of a matrix) which commutes with its conjugate transpose. (functional analysis) (Of a Hilbert space operator) which commutes with its adjoint. (category theory) (Of an epimorphism) which is the cokernel of some morphism. ...
The newpagemtimesfunction performs matrix multiplication on pages of -dimensional arrays, whilepagetransposeandpagectransposecarry out the transpose and conjugate transpose, respectively, on pages of -dimensional arrays. Performance Both releases report significantly improved speed of certain functions, includin...