Answered: Find the eigenvalues and all possible…

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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Find the eigenvalues and all possible Jordan forms if A 2 = zero matrix.

Expert Solution

Step 1

If A^2 = 0, then the characteristic polynomial of A is λ^2 = 0, which implies that the only eigenvalue of A is λ = 0.

To determine the possible Jordan forms for A, we need to examine the possible sizes and shapes of the Jordan blocks associated with the eigenvalue λ = 0. Since A^2 = 0, we know that the minimal polynomial of A divides λ^2, which means that the only possible Jordan block sizes are 1x1 and 2x2.

If A has a 1x1 Jordan block, then A is a scalar matrix with A = [0]. This is not an interesting case, as any matrix with a single 1x1 block is diagonalizable.

If A has a 2x2 Jordan block, then A is of the form:

A = [0 1]

[0 0]

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