Past exam of the mathematics course of the University of Oxford / 2021 / Prelims / Mathematics I / 1 / b / iii

Table of contents
- Solution iii

Solution

Matrix similarity transports commuting operators: with

M = P D P^{- 1}

,

CM = MC ⟺ (P^{- 1} CP) D = D (P^{- 1} CP) .

By ii, all such matrices are therefore

C = P diag (a, b, c) P^{- 1} (a, b, c \in R) .

For completeness, matrix multiplication gives

C = \frac{1}{2} a + c - b + c a - b - a + c b + c - a + b a - c b - c a + b .

The parameterization is injective, since matrix multiplication by

P^{- 1}

and

P

recovers the parameter diagonal matrix. Thus there are infinitely many choices. Already the distinct scalar matrices

t I_{3}

,

t \in R

, give an infinite family.

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