solve it ,i need urgent, no need to write neat and clean..
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2. Every 2 x 2 real matrix M = (C) determines a complex function M(x + iy) = um(X,Y) + ivm (2+iy), where real-valued functions up and um are determined by the following equation. um(2,y) UM(x,y) (a) Show that there are constants w; and w2 EC such that fm(2) = w12+ w22. What are these constants in terms of a, b,c,d? [8] (b) Determine an equivalent...
Problem 5. Let n N. The goal of this problem is to show that if two real n x n matrices are similar over C, then they are also similar over IK (a) Prove that for all X, y є Rnxn, the function f(t) det (X + ty) is a polynomial in t. (b) Prove that if X and Y are real n × n matrices such that X + ừ is an invertible complex matrix, then there exists a...
2. Let A be an n x n real symmetric matrix or a complex normal matrix. Prove that tr(A) = X1 + ... + and tr(AⓇA) = 1212 + ... +14.12 where ....... An are the eigenvalues of A repeated with multiplicity (for example, if n = 3 and the eigenvalues of A are -3 and 7 but -3 has multiplicity 2 then 11 = -3, 12 = -3, and Az = 7). 3. Let A be an n x...
Show that the real and imaginary parts of the complex-valued function f(x) = cot z are - sin 2.c sinh 2g u(I,y) v(x,y) = cos 2x - cosh 2y cos 2x - cosh 2y (cot 2 = 1/tan 2)
7. Let f:D + C be a complex variable function, write f(x) = u(x, y) +iv(x,y) where z = x +iy. (a) (9 points) (1) Present an equivalent characterization(with u and v involved) for f being analytic on D. (Just write down the theorem, you don't need to prove it.) (2) Let f(z) = (4.x2 + 5x – 4y2 + 3) +i(8xy + 5y – 1). Show that f is an entrie function. (3) For the same f as above,...
2, (15 pts) For an m × n real matrix A and vectors x and b, find ▽zllAx-bll3 and derive the normal equations.
2, (15 pts) For an m × n real matrix A and vectors x and b, find ▽zllAx-bll3 and derive the normal equations.
(2 points) Find all the eigenvalues (real and complex) of the matrix 7 5 5 M = -5 -1 -2 -5 -8 -6 The eigenvalues are (Enter your answers as a comma separated list.)
One point per regular blank unless otherwise specified, two points per blank matrix. M=CE determinant of M= trace of M= (2 points) The characteristic polynomial PM(x) = det (xl – M) = Find the two eigenvalues of M, the roots of the characteristic polynomial. 21 = – 12 = M can be decomposed into two idempotent matrices Ej and E2, with the following properties. El + E2 = 1 1E1 + 12E2 = M E Ez = [0] E2 =...
2- a) The real part of a complex function f(z) given as, u(x, y) = 3x?y - y. Iff(2) is an analytic function, find v(x,y) and f(z) (15p) b) Find the whether f(z) is analytic or not where f(z) = cos(x) +ie'sinx. (15p)
2. Consider the following set of complex 2 x 2 matrices where i = -1: H = a + bi -c+dil Ic+dia-bi Put B = {1, i, j, k} where = = {[ctdie met di]|1,3,c,dex} 1-[ ), : = [=]. ; = [i -:], « =(: :] . (a) Show that H is a subspace of the real vector space of 2 x 2 matrices with entries from C, that is, show H is closed under matrix addition and multi-...