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.

Expert Solution

Colby Messinger answered 2 years ago

(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.

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...

Construct a generator matrix and a parity check matrix for a ternary Hamming code Ham(2, 3)....

Construct a generator matrix and a parity check matrix for a ternary Hamming code Ham(2, 3). Assume a codeword x from for the ternary Hamming code Ham(2, 3)$ was sent and the word y was received. Use the partiy check matrix you constructed in part (a) to decode y in each part using syndrome decoding: (b) y = ( 1 , 1 , 1 , 0 ), (c) y = ( 2 , 2 , 2 , 2 ), (d)...

The matrix A=[-2,0,2;0,-4,0;-2,0,-6] has a single eigenvalue=-4 with algebraic multiplicity three. a.find the basis for the...

The matrix A=[-2,0,2;0,-4,0;-2,0,-6] has a single eigenvalue=-4 with algebraic multiplicity three. a.find the basis for the associated eigenspace. b.is the matrix defective? select all that apply. 1. A is not defective because the eigenvalue has algebraic multiplicity 3. 2.A is defective because it has one eigenvalue. 3.A is defective because geometric multiplicity of the eigenvalue is less than the algebraic multiplicity. 4.A is not defective because the eigenvectors are linearly dependent.

5. Compute a basis for each eigenspace of matrix A = | 6 3 −4 −1...

5. Compute a basis for each eigenspace of matrix A = | 6 3 −4 −1 | (a) Diagonalize A. (b) Compute A4 and A−3 . (c) Find a matrix B such that B2 = A. (or compute √ A.)

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%

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

A) Find Eigen values and Eigen vectors for the matrix below. A = ( 2 3...

A) Find Eigen values and Eigen vectors for the matrix below. A = ( 2 3 ; 1 5 ) this is a 2x2 matrix with 2 3 on the first row and 1 5 on the second row (B) Write down the spectral decomposition of the matrix A. (C) Is the matrix A positive definite matrix? Why?

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.

Question