determine how many Jordan canonical forms are possible with the given eigenvalues (not counting rearrangements of the Jordan blocks). You do not need to list them. Q. An 11 × 11 matrix with eigenvalues λ = 2,2,2, 2,6,6,6,6,8,8,8.
determine how many Jordan canonical forms are possible with the given eigenvalues (not counting rearrangements of the Jordan blocks). You do not need to list them.
Q. An 11 × 11 matrix with eigenvalues λ = 2,2,2, 2,6,6,6,6,8,8,8.
The number of Jordan Canonical forms are possible has to be determined.
The given eigenvalues are
The number of Jordan canonical forms that are possible for a given set of eigenvalues is equal to the number of ways the eigenvalues can be partitioned into disjoint subsets such that each subset corresponds to a single Jordan block.
For the given eigenvalues, we can partition them as follows:
Each subset corresponds to a distinct Jordan block, and the size of the block is equal to the number of eigenvalues in the subset. Therefore, the possible Jordan canonical forms are:
- Two 2x2 blocks, four 4x4 blocks, and three 3x3 blocks.
- Two 2x2 blocks, four 4x4 blocks, two 3x3 blocks, and one 1x1 block.
- Two 2x2 blocks, four 4x4 blocks, one 3x3 block, and two 1x1 blocks.
- Two 2x2 blocks, four 4x4 blocks, one 3x3 block, and one 2x2 block.
Therefore, there are four possible Jordan canonical forms for the given eigenvalues.
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