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This leads to the spectral mapping theorem: let A be a normal transformation in a complex vector space V with dim(V ) = n. Than: 1. There exist projection transformations Pi , 1 ≤ i ≤ p, with the properties • Pi · Pj = 0 for i = j, • P1 + ... + Pp = II, • dimP1 (V ) + ... , αp so that A = α1 P1 + ... + αp Pp . 2. If A is unitary than holds |αi | = 1 ∀i. 3. If A is Hermitian than αi ∈ IR ∀i. Chapter 5: Linear algebra 35 Complete systems of commuting Hermitian transformations Consider m Hermitian linear transformations Ai in a n dimensional complex inner product space V .
The transpose of A is defined by: aTij = aji . For this holds (AB)T = B T AT and (AT )−1 = (A−1 )T . For the inverse matrix holds: (A · B)−1 = B −1 · A−1 . The inverse matrix A−1 has the property that A · A−1 = II and can be found by diagonalization: (Aij |II) ∼ (II|A−1 ij ). 29 Mathematics Formulary by ir. A. , a∗n ) For the determinant det(A) of a matrix A holds: det(AB) = det(A) · det(B). , an ) + ... , ) dt dt dt dt When the rows of a matrix are considered as vectors the row rank of a matrix is the number of independent vectors in this set.
Each Lj (x) has order n, 2. , n, 3. Each polynomial p(x) can be written uniquely as n p(x) = cj Lj (x) with cj = p(xj ) j=0 This is not a suitable method to calculate the value of a ploynomial in a given point x = a. To do this, the Horner algorithm is more usable: the value s = k ck xk in x = a can be calculated as follows: float GetPolyValue(float c[], int n) { int i; float s = c[n]; for (i = n - 1; i >= 0; i--) { s = s * a + c[i]; } return s; } After it is finished s has value p(a). Mathematics Formulary by ir.