2 Lots - Ja 67 (a) Show that X(X) = 12 - T(A)+det(A), where TY(A) =...
# 2: Consider the real symmetric matrix A= 4 1 a) What are the eigenvalues and eigenvectors. [Hint: Use wolframalpha.] b) What is the trace of A, what is the sum of the eigenvalues of A. What is a general theorem th c) The eigenvalues of A are real. What is a general theorem which assert conditions that t d) Check that the eigenvectors are real. What is a general theorem which asserts conditions th asserts equality? eigenvalues are real...
(d) Show that if and are distinct eigenvalues of a square matrix A, x = (x1; x2; : : : ; xn) 2 E[A], y = (y1; y2; : : : ; yn) 2 E[A] then: x; y = x1y1 + x2y2 + + xnyn = 0:
2. Let T: P(R) + P(R) be such that Tp(x) = P(1)x2 +p(1)+ p0). a) Show that T is a linear operator. b) Find a basis for Ker(T) and a basis for Range(T). c) Is T invertible? Why? d) If possible find a basis for P(R) such that [T], is a diagonal matrix. e) Find the eigenvalues and eigenvectors of S=T* - 31.
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-...