Question

In: Math

(2) A matrix A is given. Find, if possible, an invertible matrix P and a diagonal...

(2) A matrix A is given. Find, if possible, an invertible matrix P and a diagonal matrix D such that P −1AP = D. Otherwise, explain why A is not diagonalizable.

(a) A = −3 0 −5

0 2 0

2 0 3

(b) A = 2 0 −1

1 3 −1

2 0 5

(c) A = 1 −1 2

−1 1 2

2 2 2

Expert Solution

Solution : ( 2 )

---------------------------------------------------------------------------------------------------------------------------------------------------

milcah answered 1 year ago

Diagonalize the matrix (That is, find a diagonal matrix D and an invertible matrix P such...

Diagonalize the matrix (That is, find a diagonal matrix D and an invertible matrix P such that A=PDP−1. (Do not find the inverse of P). Describe all eigenspaces of A and state the geometric and algebraic multiplicity of each eigenvalue. A= -1 3 0 -4 6 0 0 0 1

For the matrix A, find (if possible) a nonsingular matrix P such that P−1AP is diagonal....

For the matrix A, find (if possible) a nonsingular matrix P such that P−1AP is diagonal. (If not possible, enter IMPOSSIBLE.) A = 2 −2 3 0 3 −2 0 −1 2 P = Verify that P−1AP is a diagonal matrix with the eigenvalues on the main diagonal. P−1AP =

P.0.2 Show that (a) the diagonal entries of a Hermitian matrix are real; (b) the diagonal...

P.0.2 Show that (a) the diagonal entries of a Hermitian matrix are real; (b) the diagonal entries of a skew-Hermitian matrix purely imaginary; c) the diagonal entries of a skew-symmetric matrix are zero. P.0.5 Let A ∈ Mn be invertible. Use mathematical induction to prove that (A-1)k = (Ak)-1 for all integers k. P.0.25 Let A ∈ Mn be idempotent. Show that A is invertible if and only if A = I P.0.26 Let A,B ∈ Mn be idempotent. Show...

Find a matrix P that diagonalizes the matrix A = [ 2 0 ?2 / 0...

Find a matrix P that diagonalizes the matrix A = [ 2 0 ?2 / 0 3 0 / 0 0 3 ] and compute P ?1AP.

Let A be a diagonalizable n × n matrix and let P be an invertible n...

Let A be a diagonalizable n × n matrix and let P be an invertible n × n matrix such that B = P−1AP is the diagonal form of A. Prove that Ak = PBkP−1, where k is a positive integer. Use the result above to find the indicated power of A. A = 6 0 −4 7 −1 −4 6 0 −4 , A5 A5 =

Let A be a diagonalizable n × n matrix and let P be an invertible n...

Let A be a diagonalizable n × n matrix and let P be an invertible n × n matrix such that B = P−1AP is the diagonal form of A. Prove that Ak = PBkP−1, where k is a positive integer. Use the result above to find A5 A = 4 0 −4 5 −1 −4 6 0 −6

For a 2 by 2 invertible matrix A, define the condition number to be cond(A) =...

For a 2 by 2 invertible matrix A, define the condition number to be cond(A) = ||A|| ⋅ ||A||-1. Assume that the matrix norm is defined using the Euclidean vector norm. (a) Find two 2by2 invertible matrices B and C such that cond(B + C) < cond(B) + cond(C). (b) Find two 2by2 invertible matrices B and C such that cond(B + C) > cond(B) + cond(C). (c) Suppose that A is a symmetric invertible 2by2 matrix. Find cond(2A) and...

Determine if the matrix below is invertible. Use as few calculations as possible. Justify your answer....

Determine if the matrix below is invertible. Use as few calculations as possible. Justify your answer. 0 2 −4 1 0 2 −2 −6 8 Choose the correct answer below. A. The matrix is invertible. The columns of the given matrix span are linearly dependent. B. The matrix is not invertible. If the given matrix is A, the equation Ax=b has a solution for at least one b in set of real numbers ℝ3. C.The matrix is invertible. The given...

Exercise 2.1.39 Let A be a 2×2 invertible matrix, with A = [a b c d]...

Exercise 2.1.39 Let A be a 2×2 invertible matrix, with A = [a b c d] Find a formula for A−1 in terms of a,b, c,d by using elementary row operations

Consider an orthogonal matrix Q ∈ R ^m×m and a diagonal matrix with non-zero diagonal elements...

Consider an orthogonal matrix Q ∈ R ^m×m and a diagonal matrix with non-zero diagonal elements D ∈ R^ m×m and answer the True— False questions below. (k) Any vector x ∈ R^m can be written as a linear combination of the columns of D (l) The columns of Q span R^m (m) The columns of D span R^m (n) The columns of Q are a basis for R^m (o) The columns of D are a basis for R^m

Question

(2) A matrix A is given. Find, if possible, an invertible matrix P and a diagonal...

Solutions

Expert Solution

Related Solutions

Diagonalize the matrix (That is, find a diagonal matrix D and an invertible matrix P such...

For the matrix A, find (if possible) a nonsingular matrix P such that P−1AP is diagonal....

P.0.2 Show that (a) the diagonal entries of a Hermitian matrix are real; (b) the diagonal...

Find a matrix P that diagonalizes the matrix A = [ 2 0 ?2 / 0...

Let A be a diagonalizable n × n matrix and let P be an invertible n...

Let A be a diagonalizable n × n matrix and let P be an invertible n...

For a 2 by 2 invertible matrix A, define the condition number to be cond(A) =...

Determine if the matrix below is invertible. Use as few calculations as possible. Justify your answer....

Exercise 2.1.39 Let A be a 2×2 invertible matrix, with A = [a b c d]...

Consider an orthogonal matrix Q ∈ R ^m×m and a diagonal matrix with non-zero diagonal elements...