Complex conjugate transposed

Note that the Liouville matrix, iL+R+K may not be Hennitian, but it can still be diagonalized. Its eigenvalues and eigenvectors are not necessarily real, however, and the inverse of U may not be its complex-conjugate transpose. If complex numbers are allowed in it, equation (B2.4.33) is a general result. Since A is a diagonal matrix it can be expanded in tenns of the individual eigenvalues, X. . The inverse matrix can be applied... 

The functions tpi(x) are, in general, complex functions. As a consequence, ket space is a complex vector space, making it mathematically necessary to introduce a corresponding set of vectors which are the adjoints of the ket vectors. The adjoint (sometimes also called the complex conjugate transpose) of a complex vector is the generalization of the complex conjugate of a complex number. In Dirac notation these adjoint vectors are called bra vectors or bras and are denoted by or (/. Thus, the bra (0,j is the adjoint of the ket, ) and, conversely, the ket j, ) is the adjoint (0,j of the bra (0,j... 

Here UT denotes the transpose matrix, namely if U = (uiyj) is given by its entries, then UT = (Ujyi) with rows and columns exchanged, and U = (ujyi) is the complex conjugate matrix with complex conjugate transposed entries. 

Chapter 11, matrix P describes a rotation, such as the result of a 90° pulse. In such instances matrix P is found to be unitary and to have a simple inverse, the complex conjugate transpose—(P 1)m = P m —in which rows and columns are interchanged and each i changes to — i. For example,... 

Left or right multiplication by A thus turns the matrix A into the unit matrix, and the inverse of a matrix is unique it thus follows that the inverse of A is obtained by taking the complex-conjugate transposed form, which means that the matrix is unitary. Note that the result in Eq. (2.22) is valid only if the matrix A is nonsingular. However, this will certainly be the case since... 

A matrix H which is equal to its complex-conjugate transpose, H = is called Hemitian. It follows that the diagonal elements of such a matrix are real, while corresponding off-diagonal elements form complex-conjugate pairs ... 

In order to apply the transformation to the interaction operator, we must also consider the effect of R on the column of bra-functions. This simply requires the inverse of the matrix, which, for a unitary matrix, is nothing but its complex conjugate transposed ... 

B Complex Conjugate, Transpose, and Hermitian Adjoint of a Matrix... 

The complex conjugate transpose of a matrix is called the adjoint and is designated with a superscript t. Thus, A+ = AT ... 

The overlap of the F function with itself is the dot product of the coefficient vector and its complex conjugate transpose if, as we assume here, the basis functions are orthonormal. 

Optical elements can be represented by 2 x 2 matrices, wbicb act from the left by matrix multiplication on J to produce a new vector representing the light leaving the element. The leading phase term e" is not needed to describe relative phase shifts and is usually dropped for simplicity. From this description, the intensity of light is calculated by taking tbe modulus of tbe complex inner product of the Jones vector I = J J, where is the complex-conjugate transpose. 

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