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Chapter 5: Linear algebra 39 Hermitian transformations A transformation H : V → V with V = C n is Hermitian if (Hx, y ) = (x, Hy ). The Hermitian conjugated transformation AH of A is: [aij ]H = [a∗ji ]. An alternative notation is: AH = A† . The inner product of two vectors x and y can now be written in the form: (x, y ) = xH y. If the transformations A and B are Hermitian, than their product AB is Hermitian if: [A, B] = AB − BA = 0. [A, B] is called the commutator of A and B. The eigenvalues of a Hermitian transformation belong to IR.

The set vectors {an } is linear independent if: λi ai = 0 ⇔ ∀i λi = 0 i The set {an } is a basis if it is 1. independent and 2. V =< a1 , a2 , ... 1 Basic operations λi ai . For the matrix multiplication of matrices A = aij and B = bkl holds with r the row index and k the column index: Ar1 k1 · B r2 k2 = C r1 k2 , (AB)ij = aik bkj k 33 34 Mathematics Formulary by ir. A. Wevers where r is the number of rows and k the number of columns. The transpose of A is defined by: aTij = aji . For this holds (AB)T = B T AT and (AT )−1 = (A−1 )T .

Define f (a) = c0 and the series is also valid for |z − a| < R and f is analytical in a. 2. The principal part contains a finite number of terms. Than there exists a k ∈ IN so that lim (z − a)k f (z) = c−k = 0. Than the function g(z) = (z − a)k f (z) has a non-essential singularity in z→a a. One speaks of a pole of order k in z = a. 3. The principal part contains an infinite number of terms. Then, a is an essential singular point of f , such as exp(1/z) for z = 0. e. 5 49 Jordan’s theorem Residues are often used when solving definite integrals.

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