Conjugated matrix

The operation of complex conjugation will be denoted by an overscore a denotes the complex conjugate of a. For a matrix A, with matrix elements al the hermitian conjugate matrix with elements afl will be denoted 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. 

Roots of a self-conjugated matrix are always real (this is a well-known fact see, for example, ref. 29). To prove negative determinacy L in the subspace generated by vectors ySja, it is necessary and sufficient to show that < ysa Lysa < 0 at any s or a. We obtain... 

The transpose of the complex conjugate matrix (sometimes termed the adjoint matrix), is written as A and defined such that ... 

A Q to obtain a conjugate matrix 6 that has been block-diagonalized, as shown by Equation (8.19). You should do the matrix multiplication yourself to prove that the similarity transform converts our original matrix into a diagonalized matrix. It is not important for you to understand where Q and Q came from ... 

For this reason, the matrix in Eq. (7.37) is also called the conjugating matrix. The conjugating matrix is defined up to an arbitrary phase. We have taken here the standard phase choice. The conjugating relationships between the two spins can now be used to transfer the spin functions in the coupling coefficients from the ket to the bra part. An a spin in the ket part becomes a spin in the bra part, while a spin in the ket becomes a -a spin in the bra part. 

The transposed conjugate matrix is the inverse matrix for unitary matrices. P is thus a unit matrix with 1 in the diagonal and otherwise zeroes. 

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