 Solution

ID: past-exam-of-the-year-1-of-the-computer-science-course-of-the-university-of-oxford/2022/digital-systems-linear-algebra-introduction-to-formal-proof/4/b/solution

Let

P A P^{- 1} = diag (α_{1}, \dots, α_{n})

and

PB P^{- 1} = diag (β_{1}, \dots, β_{n})

. Then

P (A + B) P^{- 1} = diag (α_{1} + β_{1}, \dots, α_{n} + β_{n}) .

By matrix similarity, every eigenvalue of $A + B$ is $α_{i} + β_{i}$ for some $i$ . These are corresponding eigenvalues on the common eigenbasis, rather than every possible pairwise sum. This is eigenvalue addition in a common eigenbasis.

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