Derive eigenvalues and sum for inverse matrix
Eigenvalues of an Inverse and Their Sum
Context
Let A be an invertible n×n matrix (over the real or complex numbers). All eigenvalues of A are nonzero because A is invertible.
Tasks
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Prove that the eigenvalues of A^{-1} are {1/λ1, …, 1/λn}, where {λ1, …, λn} are the eigenvalues of A (counted with algebraic multiplicity).
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Compute the sum of the eigenvalues of A^{-1} and express it in terms of A (e.g., as tr(A^{-1})).
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State any assumptions needed for these equalities to hold.
Constraints & Assumptions
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Preserve the scope, facts, inputs, and requested outputs from the prompt above.
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If the prompt leaves a detail unspecified, state a reasonable assumption before relying on it.
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Keep the answer interview-ready: concise enough to present, but concrete enough to implement or evaluate.
Clarifying Questions to Ask
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Clarify the random variables, distributional assumptions, independence assumptions, and desired output.
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Show enough derivation for the interviewer to follow the reasoning.
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Explain how you would validate the result with simulation or sensitivity checks.
What a Strong Answer Covers
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A correct setup with definitions, formulas, and boundary conditions.
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A step-by-step derivation or estimation plan.
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Interpretation of the result, including uncertainty and practical limitations.
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Checks for assumptions, edge cases, and numerical stability.
Follow-up Questions
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How would the result change if the assumptions were relaxed?
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Can you verify the answer with a simulation?
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What is the most likely source of estimation error?