In this question, you will build a matrix from the eigenvectors and eigenvalues, instead of the other way around. The matrix A has three eigenvalues: A₁ = -4 with eigenvector ₁ = (1,1,0), A2 = 3 with eigenvector u2 = (1,-1,1) and X3 = 12 with eigenvector u3 = (1,-1,-2). (a) Normalize the eigenvectors u; to give vi. Enter them in the usual format e.g. [1,2,3]. 5 v1 = v2 = v3 = (b) Recall that you can build an orthogonal matrix P whose columns are that set of orthonormal eigenvectors. Then PT AP = D, where D is a diagonal matrix that contains the eigenvalues along the diagonal. We now have P and D, so can find A from the formula A = PDPT Enter the matrix A as a list of row vectors.
In this question, you will build a matrix from the eigenvectors and eigenvalues, instead of the other way around. The matrix A has three eigenvalues: A₁ = -4 with eigenvector ₁ = (1,1,0), A2 = 3 with eigenvector u2 = (1,-1,1) and X3 = 12 with eigenvector u3 = (1,-1,-2). (a) Normalize the eigenvectors u; to give vi. Enter them in the usual format e.g. [1,2,3]. 5 v1 = v2 = v3 = (b) Recall that you can build an orthogonal matrix P whose columns are that set of orthonormal eigenvectors. Then PT AP = D, where D is a diagonal matrix that contains the eigenvalues along the diagonal. We now have P and D, so can find A from the formula A = PDPT Enter the matrix A as a list of row vectors.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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![In this question, you will build a matrix from the eigenvectors and eigenvalues, instead of the other way around.
The matrix A has three eigenvalues:
A₁ = -4 with eigenvector ₁ = (1,1,0),
A2 = 3 with eigenvector u2 = (1,-1,1) and
X3 = 12 with eigenvector u3 = (1,-1,-2).
(a) Normalize the eigenvectors u; to give vi. Enter them in the usual format e.g. [1,2,3].
5
=
v1 =
v2 =
v3
=
(b) Recall that you can build an orthogonal matrix P whose columns are that set of orthonormal eigenvectors.
Then PT AP = D, where D is a diagonal matrix that contains the eigenvalues along the diagonal.
We now have P and D, so can find A from the formula A = PDPT Enter the matrix A as a list of row vectors.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F4ac09755-9639-4075-b6f1-d2ae73f82d81%2Fe3c69f57-e71b-4165-a9a4-d6b37ef2f6ec%2Filvlcve_processed.png&w=3840&q=75)
Transcribed Image Text:In this question, you will build a matrix from the eigenvectors and eigenvalues, instead of the other way around.
The matrix A has three eigenvalues:
A₁ = -4 with eigenvector ₁ = (1,1,0),
A2 = 3 with eigenvector u2 = (1,-1,1) and
X3 = 12 with eigenvector u3 = (1,-1,-2).
(a) Normalize the eigenvectors u; to give vi. Enter them in the usual format e.g. [1,2,3].
5
=
v1 =
v2 =
v3
=
(b) Recall that you can build an orthogonal matrix P whose columns are that set of orthonormal eigenvectors.
Then PT AP = D, where D is a diagonal matrix that contains the eigenvalues along the diagonal.
We now have P and D, so can find A from the formula A = PDPT Enter the matrix A as a list of row vectors.
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