Problem 3. Throughout this problem, we fix a matrix A ∈ Fn,n with the property that...

Problem 3. Throughout this problem, we fix a matrix A ∈ Fn,n with the property that A = A∗. (If F = R, then A is called symmetric. If F = C, then A is called Hermitian.) For u, v ∈ Fn,1, define [u, v] = v∗ Au. (a) Let Show that K is a subspace of Fn,1. K:={u∈Fn,1 :[u,v]=0forallv∈Fn,1}. (b) Suppose X is a subspace of Fn,1 with the property that [v,v] > 0 for all nonzero v ∈ X. (1) Show that [−, −] defines an inner product on X. (c) Suppose Y is a subspace of Fn,1 with the property that [v,v] < 0 for all nonzero v ∈ Y. (2) If X is a subspace with property (1), prove that X + K + Y is a direct sum, where K is defined in part (a).

Solutions

Expert Solution

Colby Messinger answered 1 year ago

Next > < Previous

Related Solutions

Suppose A is an n × n matrix with the property that the equation Ax =...

Suppose A is an n × n matrix with the property that the equation Ax = b has at least one solution for each b in R n . Explain why each equation Ax = b has in fact exactly one solution

Let f1 = 1 and f2=1 and for all n>2 Let fn = fn-1+fn-2. Prove that...

Let f1 = 1 and f2=1 and for all n>2 Let fn = fn-1+fn-2. Prove that for all n, there is no prime p that divides noth fn and fn+1

Problem 1 1.1 If A is an n x n matrix, prove that if A has...

Problem 1 1.1 If A is an n x n matrix, prove that if A has n linearly independent eigenvalues, then AT is diagonalizable. 1.2 Diagonalize the matrix below with eigenvalues equal to -1 and 5. 0 1 1 2 1 2 3 3 2 1.3 Assume that A is 4 x 4 and has three different eigenvalues, if one of the eigenspaces is dimension 1 while the other is dimension 2, can A be undiagonalizable? Explain. Answer for all...

What property must a symmetric 3 × 3 matrix have in order for the equation xTAx...

What property must a symmetric 3 × 3 matrix have in order for the equation xTAx = 1 to represent an ellipsoid? (6 points)(Kindly provide a long, comprehensive proof)

Suppose C is a m × n matrix and A is a n × m matrix....

Suppose C is a m × n matrix and A is a n × m matrix. Assume CA = Im (Im is the m × m identity matrix). Consider the n × m system Ax = b. 1. Show that if this system is consistent then the solution is unique. 2. If C = [0 ?5 1 3 0 ?1] and A = [2 ?3 1 ?2 6 10] ,, find x (if it exists) when (a) b =[1...

Is police brutality in the U.S a problem? If so, how do Do We fix it?

give a constructive proof of fn = Q^n + P^n/ Q - P , where Q...

give a constructive proof of fn = Q^n + P^n/ Q - P , where Q is the positive root and P is negative root of x^2 - x - 1= 0 fn is nth term of fibonacci sequence, f1 = 1 f2, f3 = f2 +f1, ... fn= fn_1 +fn_2 , n>2

Find the pointwise limit f(x) of the sequence of functions fn(x) = x^n/(n+x^n) on [0, ∞)....

Find the pointwise limit f(x) of the sequence of functions fn(x) = x^n/(n+x^n) on [0, ∞). Explain why this sequence does not converge to f uniformly on [0,∞). Given a > 1, show that this sequence converges uniformly on the intervals [0, 1] and [a,∞) for any a > 1.

Matrix A belongs to an n×n matrix over F. show that there exists a nonzero polynomial...

Matrix A belongs to an n×n matrix over F. show that there exists a nonzero polynomial f(x) belongs to F[x] such that f(A) ＝0.

True or False? Why? Σ n = 1, ∞ fn(x) converges uniformly on A <=> for...

True or False? Why? Σ n = 1, ∞ fn(x) converges uniformly on A <=> for all n in N (natural numbers), there exists Mn > 0 such that |fn(x)| <= Mn for all x in A and Σ n = 1, ∞ Mn converges.

Subjects