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.

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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.

Expert Solution
Step 1: Given Information:

The number of Jordan Canonical forms are possible has to be determined.

The given eigenvalues are  λ = 2,2,2, 2,6,6,6,6,8,8,8.

Step 2: Calculation

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:

2  2 2  2  6 6  6 6  8 8 8

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:

  1. Two 2x2 blocks, four 4x4 blocks, and three 3x3 blocks.
  2. Two 2x2 blocks, four 4x4 blocks, two 3x3 blocks, and one 1x1 block.
  3. Two 2x2 blocks, four 4x4 blocks, one 3x3 block, and two 1x1 blocks.
  4. 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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