For each matrix A, write A = PJP^(−1) with P an invertible matrix and J a matrix in Jordan Canonical Form?
For each matrix A, write A = PJP^(−1) with P an invertible matrix and J a matrix in Jordan Canonical Form
A= | 3 0 2 |
……| 1 3 1 |
……| 0 1 1 |
det(λI − A) = (λ − 1)(λ − 2)(λ − 4).
Does anyone could help me with this exercise please?
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Since the matrix has distinct eigenvalues λ = 1, 2, 4, we see that the JCF for A is diagonal.
Computing the eigenvectors:
(i) For λ = 1, we row reduce (A - 1I)v = 0.
[2 0 2|0]
[1 2 1|0]
[0 1 0|0], which row reduces to
[1 0 1|0]
[0 1 0|0]
[0 0 0|0], yielding eigenvector (-1, 0, 1)^t.
(ii) For λ = 2, we row reduce (A - 2I)v = 0.
[1 0 2|0]
[1 1 1|0]
[0 1 -1|0], which row reduces to
[1 0 2|0]
[0 1 -1|0]
[0 0 0|0], yielding eigenvector (-2, 1, 1)^t.
(iii) For λ = 4, we row reduce (A - 4I)v = 0.
[-1 0 2|0]
[1 -1 1|0]
[0 1 -3|0], which row reduces to
[1 0 -2|0]
[0 1 -3|0]
[0 0 0|0], yielding eigenvector (2, 3, 1)^t.
So, setting J =
[1 0 0]
[0 2 0]
[0 0 4], we have
P =
[-1 -2 2]
[0 1 3]
[1 1 1].
I hope this helps!