Let A be the real 2x2 matrix [a b] [c d] with bc greater than or equal to 0. Prove there exists a real 2x2 invertible matrix S so that S^-1 A S is either diagonal or of the form [x 1] [0 x] where x is the eigenvalue of A.
Let B be an nxn matrix with B^2 = B prove that B is diagonalizable, ie there exists an invertible matrix S so that S^-1 B S is diagonal. (Hint: all eigenvalues of B are either 0 or 1. For each k between 0 and n, consider the case when the nullity of B is k.)
Suppose A & B are Hermitian matrices and AB=BA, show that A and B are simultaneously diagonalizable, ie, there exists an unitary matrix C so that both C*AC adn C*BC are diagonal.
1) If a Matrix A is diagonizable, must it have an inverse ? if so, is it diagonizable? Can {see attachment} be diagonized, does it have an inverse as well as {see attachment} 2) A is mxn For m<n, is there a vector b such that Ax = b does not have any solution? Any trivial solution for Ax = 0? b) Can say the same for m>n ? A
Given y " + ky = 0; y(0)=0 and y'(1)=0; a) Determine the normalized eigenfunction for this problem; b) Use the results in part (a) to express f(x)=x in an eigenfunction expansion, i.e. determine the expansion coefficients (Cn).
Calculate the eigenvalues of this matrix: -16 6 60 2 [Note-- you'll probably want to use a graphing calculator to estimate the roots of the polynomial which defines the eigenvalues..... (see attached)]
Calculate the eigenvalues of this matrix: {see attachment} Note: You'll probably want to use a graphing calculator to estimate the roots of the polynomial which defines the eigenvalues.
Find the general real-valued solution of each system. Classify the origin as a saddle, center, spiral, or one of the normal types (identify the type). Identify as neutrally stable, unstable, or asymptotically stable. *See attachment for systems
LET F be a field and set G = a b -b a : a,b is an element of F. Under what conditions on F will G be a field? Can you give an example of such F other than R (real numbers)?
Suppose that the initially lead-free subject in Example 6.1.1 is exposed to lead for 400 days, and then removed to a lead free environment. Use a computer to estimate how long it takes the amount x3 of lead in the bones to decline 50% of x3(400); repeat for 25% and 10%. (See attachment for full details)
The matrix A = 1 1 0 0 0 0 0 1 1 has two real eigenvalues, one of multiplicity 1 and one of multiplicity 2. Find the eigenvalues and the eigenvectors. Eigenvalue of multiplicity 1 : 0 Associated Eigenvector: .57735 -.57735 .57735
Let A = [ -5, 6, -19] B= [-1, 2, -3] and C= [-2, 2, -8] **they are linearly independent*** The problem is the same as if the vectors were written vertically. If they are linearly dependent, determine a non-trivial linear relation - (a non-trivial relation is three numbers which are not all three zero.) otherwise, if t
The matrix A = 1 1 0 0 0 0 0 1 1 has two real eigenvalues, one of multiplicity 1 and one of multiplicity 2. Find the eigenvalues and the eigenvectors. Eigenvalue of multiplicity 1 : Associated Eigenvector: Eigenvalue of multiplicity 2 : Associated two linearly independent
For which value of k does the matrix A = 1 k 4 8 have one real eigenvalue of multiplicity 2? K=___?
A = -14 -4 20 4 smaller eigenvalue = -6 associated eigenvector= (__ , __) larger eigenvalue = -4 associated eigenvector= (__ , __) Find associated eigenvectors.
Matrix A (-4k) has two distinct real eigenvalues if and only if k > ____? (19 ). See the attached file.
Prove that (n,m) denotes the greatest common devisor of m and n See Attached
Verify: (a) If A is diagonalizable and B is similar to A then B is also diagonalizable. (b) If {see attachment} and x is an eigenvector of A corresponding to an eigenvalue ... {see attachment for complete question
? Let G be a group and let a,b be two elements of G. The conjugate of b by a is, by definition, the element . The centralizer of a, denoted by s the set of all elements g in G such that ga=ag. i) Find all possible conjugates f the permutation ii) Find the centralizer p in . iii) Prove that for any element a in a g
1. Solve by substitution or elimination method: 3x - 2y = 8 -12x + 8y = 32 2. Solve by substitution or elimination method: 7x - 5y = 14 -4x + y = 27 3. Solve by substitution or elimination method: -4x + 3y = 5 12x - 9y = -15 4. A university boo
1. Why do intersecting lines represent a unique solution? Give examples to support your answer. 2. What is the significance of the name 'linear equation' to its graphical representation? 3. The solutions of line m are (3, 9), (5, 13), (15, 33), (34, 71), (678, 1359), and (1234, 2471). The solutions of line n are (3, -9)
1.Can you show that, given two equations y = m1x + c1 and y = m2x + c2 where c1 and c2 are different, there is no solution if m1 = m2. Interpret this result graphically. Also show that if c1 = c2 then there will be at least one solution no matter what m1 and m2 are. Interpret this result on a graph. 2.In your reading you have
Please show all the steps involved. 1. An nxn matrix A is said to be nilpotent if A^k = O (the zero matrix) for some positive integer k. Show that all the eigen values of a nilpotent matrix are O.
Please see the attached file for the fully formatted problems. Find a second linearly independent solution given the differential equation & non-trivial solution f.
Only problems #3 &4-a,(without using any software). 3 . Ax = b we consider the iterative scheme .... where the matrix Q is nonsingular. (a) If ... for some subordinate matrix norm, show that the sequence produced by the above scheme converges to the solution of the system for any initial vector x(0). 4. Given singular
One solution to ty"-(t+2)y'+2y=0 is exp(t) Find a second linearly independent solution.
3. (a) Solve the following systems of equations i) x + 2 y - z = 2 -3 x - y + z = -3 - x+ 3 y - z = 1 ii) 4 x+ -3 y+ z = - 1 -3 x+ y+ -5 z = 0 -5 x -4 z = 0 iii) x1 + x2 + x3 = 3 -3 x1 -17 x2 + x3 + 2 x 4 = 1 4 x -7 x2 + 8 x3 -5 x4 = 1 -5 x2 -2 x3 + x4 = 1 (b) Find the values of k for which
Find the fixed points and sketch trajectories in the phase plane for the system: ... using the phase portrait, examine the behaviour of solutions of this system as t→∞ when they start from (x,y)=(-1,0), and when they start from (x,y)=(-1,-1). Please see attached for full question.
If A={(-1,-1),(3,-4),(-2,5),(0,3),(2,1),(4,7)}what would be the convexhull(A) expressed as the intersection of a minimum number of closed halfplanes. ALSO, if K is the intersection of the halfspaces: {(x,y,z):x>=0} {(x,y,z):y>=0} {(x,y,z):z>=0} {(x,y,z):x+2y+3z<=6} {(x,y,z):x+3y+2z<=6} {(x,y,z):x<=4} what would the ver
True or false? Justify your answer in each case (giving a proof or a counterexample): Let T:V-->W be a linear transformation which is an isomorphism. Denote its inverse by T-1 . Suppose that (SYMBOL) is an eigenvalue of T. Then answer a, b, c and d. PLEASE SEE ATTACHMENT FOR COMPLETE QUESTION