Past exam of the year 1 of the computer science course of the University of Oxford / 2023 / Digital systems, linear algebra and introduction to formal proof / 6 / d

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- Solution d

Solution

If

A^{⊤} = - A

, then the dot product satisfies

(A x) \cdot y = x^{⊤} A^{⊤} y = - x^{⊤} A y = - (A y) \cdot x .

Conversely, the assumed dot product identity gives

x^{⊤} (A^{⊤} + A) y = 0

for every

x, y

. Take

x = e_{i}

and

y = e_{j}

from the standard basis: every entry of

A^{⊤} + A

is zero. Hence $A^{⊤} = - A$ , proving the equivalence.

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