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Past exam of the mathematics course of the University of Oxford

2021

Prelims

Mathematics I

{c}

<Matrix similarity> transports <commuting operators>: with $M=PDP^{-1}$,
$$
CM=MC\quad\Longleftrightarrow\quad
(P^{-1}CP)D=D(P^{-1}CP).
$$
By <1/b/ii>, all such <matrices> are therefore
$$
\boxed{C=P\operatorname{diag}(a,b,c)P^{-1}\quad(a,b,c\in\mathbb R).}
$$
For completeness, <matrix multiplication> gives
$$
C=\frac12\begin{pmatrix}
a+c&-a+c&a-c\\
-b+c&b+c&b-c\\
a-b&-a+b&a+b
\end{pmatrix}.
$$
The parameterization is <injective>, since <matrix multiplication> by $P^{-1}$ and $P$ recovers the parameter <diagonal matrix>. Thus there are \b[infinitely many] choices. Already the distinct <scalar matrices> $tI_3$, $t\in\mathbb R$, give an infinite family.

Solution

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

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

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