 Solution

ID: past-exam-of-the-linear-algebra-course-of-the-university-of-oxford/2023/2/b/solution

Δ_{i} = d_{i} - d_{i - 1}

are positive integers by part (a)(i), are nonincreasing by part (a)(ii), and sum to

d_{m} = 4

. Thus they form an integer partition of

4

. The complete list is

(4)

(3, 1)

(2, 2)

(2, 1, 1)

, and

(1, 1, 1, 1)

To prove all five occur over every field, let

J_{s} (0)

s

. It sends each standard basis vector to its predecessor, with the first sent to zero, so

dim ker J_{s} (0)^{i} = min (i, s) .

For a block diagonal matrix, the nullities add. The following explicit nilpotent matrices therefore realize every possible sequence:

m 12234 (d_{1}, \dots, d_{m}) (4) (3, 4) (2, 4) (2, 3, 4) (1, 2, 3, 4) realizing matrix 0_{4 \times 4} J_{2} (0) \oplus J_{1} (0) \oplus J_{1} (0) J_{2} (0) \oplus J_{2} (0) J_{3} (0) \oplus J_{1} (0) J_{4} (0) .

In each row the largest Jordan block size is exactly the nilpotency index

m

, so the displayed sequence stops at the correct place.

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