Let ? be an eigenvalue of a matrix A. Explain why dim(?) ? 1

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milcah answered 1 year ago

4. Let A be the 6*6 diagonal matrix below. For each eigenvalue, compute the multiplicity of...

4. Let A be the 6*6 diagonal matrix below. For each eigenvalue, compute the multiplicity of λ as a root of the characteristic polynomial and compare it to the dimension of the eigenspace Eλ. (x 0 0 0 0 0 0 x 0 0 0 0 0 0 y 0 0 0 0 0 0 x 0 0 0 0 0 0 z 0 0 0 0 0 0 x) 5. Let A be an 3*3 upper triangular matrix with...

1. What is the definition of an eigenvalue and eigenvector of a matrix? 2. Consider the...

1. What is the definition of an eigenvalue and eigenvector of a matrix? 2. Consider the nonhomogeneous equationy′′(t) +y′(t)−6y(t) = 6e2t. (a)Find the general solution yh(t)of the corresponding homogeneous problem. (b)Find any particular solution yp(t)of the nonhomogeneous problem using the method of undetermined Coefficients. c)Find any particular solution yp(t)of the nonhomogeneous problem using the method of variation of Parameters. (d) What is the general solution of the differential equation? 3. Consider the nonhomogeneous equationy′′(t) + 9y(t) =9cos(3t). (a)Findt he general...

Let U be a subspace of V . Prove that dim U ⊥ = dim V...

Let U be a subspace of V . Prove that dim U ⊥ = dim V −dim U.

Which of the following statements is/are true? 1) If a matrix has 0 as an eigenvalue,...

Which of the following statements is/are true? 1) If a matrix has 0 as an eigenvalue, then it is not invertible. 2) A matrix with its entries as real numbers cannot have a non-real eigenvalue. 3) Any nonzero vector will serve as an eigenvector for the identity matrix.

(a) Find a 3×3 matrix A such that 0 is the only eigenvalue of A, and...

(a) Find a 3×3 matrix A such that 0 is the only eigenvalue of A, and the space of eigenvectors of 0 has dimension 1. (Hint: upper triangular matrices are your friend!) (b) Find the general solution to x' = Ax. PLEASE SHOW YOUR WORK CLEARLY.

The eigenvalues of the coefficient matrix can be found by inspection or factoring. Apply the eigenvalue...

The eigenvalues of the coefficient matrix can be found by inspection or factoring. Apply the eigenvalue method to find a general solution of the system. x'1 = 6x1 + 6x2 + 2x3 x'2 = -8x1 - 8x2 - 6x3 x'3 = 8x1 + 8x2 + 6x3 What is the general solution in matrix form? x(t) =

Let A = 2 0 1 0 2 0 1 0 2 and eigenvalue λ1 =...

Let A = 2 0 1 0 2 0 1 0 2 and eigenvalue λ1 = 3 and associated eigenvector v(1) = (1, 0, 1)t . Find the second dominant eigenvalue λ2 (or the approximation to λ2) by the Wielandt Dflation method

Explain why the functions diag, dim, length, and names can be as- signed new values (as...

Explain why the functions diag, dim, length, and names can be as- signed new values (as in diag(m)=pi). Explain why (or why not) they are useful.

Estimate the lowest eigenvalue pair of matrix A using the Inverse Power Method A = 2 8...

Estimate the lowest eigenvalue pair of matrix A using the Inverse Power Method A = 2 8 10 8 4 5 10 5 7 starting with initial guess x0 = [1 1 1]T and εaλ ≤ 1%

2. For each 3*3 matrix and each eigenvalue below construct a basis for the eigenspace Eλ....

2. For each 3*3 matrix and each eigenvalue below construct a basis for the eigenspace Eλ. A= (9 42 -30 -4 -25 20 -4 -28 23), λ = 1,3 A= (2 -27 18 0 -7 6 0 -9 8), λ = −1,2 3. Construct a 2×2 matrix with eigenvectors(4 3) and (−3 −2) with eigen-values 2 and −3, respectively.

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