Conjugate transpose

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... 

Here tp denotes the conjugate transpose of ip. Another conserved quantity is the norm of the vector ip, i.e., ip ip = const, due to the unitary propagation of the quantum part. 

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... 

These are the n x n matrices with complex entries such that the inverse is the conjugate transpose. That is... 

Example 15.4. Determine the conjugate transpose of the complex matrix C ... 

The singular values of a complex matrix are similar to those of a reof matrix. The only difference is that we use the conjugate transpose. 

First we calculate the conjugate transpose [see Eq. (15.9)]. Then we multiply 4 by it. Then we calculate the eigenvalues. These can be found using Eq. (15.18) for simple systems. In more realistic problems the IMSL subroutine EIGCC can... 

MADD adds complex matrices 4 nd to give C MMULT multiplies complex matrices and to give C IDENT forms an N x Af identity matrix I CONJT takes the conjugate transpose of matrix ... 

We denote the set of complex numbers by C. Readers should be familiar with complex numbers and how to add and multiply them, as described in many standard calculus texts. We use i to denote the square root of —1 and an asterisk to denote complex conjugation if x and y are real numbers, then (x + iy) = X — iy. Later in the text, we will use the asterisk to denote the conjugate transpose of a matrix with complex entries. This is perfectly consistent if one thinks of a complex number x + iy as a one-by-one complex matrix ( x + iy ). See also Exercise 1.6. The absolute value of a complex number, also known as the modulus, is denoted... 

Then define T to be the linear transformation from VP to V whose matrix in the given basis is the conjugate transpose A of A for each j and k wq have matrix entries... 

Note that on a finite-dimensional vector space V, a linear operator is Her-mitian if and only if T = T. More concretely, in C", a linear operator is Hermitian-symmetric if and only if its matrix M in the standard basis satis-lies M = M, where M denotes the conjugate transpose matrix. To check that a hnear operator is Hermitian, it suffices to check Equation 3.2 on basis vectors. Physics textbooks often contain expressions such as (+z H — z). These expressions are well defined only if H is a Hermitian operator. If H yNQK not Hermitian, the value of the expression would depend on where one applies the H. 

Exercise 3.25 Suppose M is an nystrictly positive. Define... 

Note that every matrix AT of 5 is Hermitian symmetric, i.e., writing M to denote the conjugate transpose of M, v/e have M = M. Note also that the trace of each M e 5 is zero and... 

Another way to make the dual space (C ) concrete is to use the complex scalar product and think of elements of the dual space as column vectors. Recall the notation for the conjugate transpose of a vector. In this interpretation... 

The matrix obtained by taking the complex conjugate of each element of A and then forming the transpose is called the Hermitian conjugate (or conjugate transpose) of A and is symbolized by A" ... 

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. 

We saw that for the inverse of a matrix, A-1 A = A A 1 = 1, so for an orthogonal matrix ATA = AAT = 1, since here the transpose is the inverse. Check this out for the matrices shown. The complex analogue of an orthogonal matrix is a unitary matrix its inverse is its conjugate transpose. 

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,... 

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