Derive eigenvalues and sum for inverse matrix

Quick Overview

Derive eigenvalues and sum for inverse matrix evaluates statistical assumptions, formulas, estimation strategy, uncertainty, edge cases, and interpretation in a realistic interview setting. A strong answer states assumptions, handles edge cases, explains trade-offs, and shows how to validate the result clearly.

Derive eigenvalues and sum for inverse matrix

Company: DRW

Role: Data Scientist

Category: Statistics & Math

Difficulty: easy

Interview Round: Onsite

Let A be an invertible n×n matrix with eigenvalues {λ1, …, λn} (all nonzero). Prove that the eigenvalues of A^{-1} are {1/λ1, …, 1/λn}. Then compute the sum of the eigenvalues of A^{-1} and express it in terms of A (e.g., as tr(A^{-1})); discuss any assumptions needed for these equalities to hold.

Quick Answer: Derive eigenvalues and sum for inverse matrix evaluates statistical assumptions, formulas, estimation strategy, uncertainty, edge cases, and interpretation in a realistic interview setting. A strong answer states assumptions, handles edge cases, explains trade-offs, and shows how to validate the result clearly.

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Jul 28, 2025, 12:00 AM
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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

  1. Prove that the eigenvalues of A^{-1} are {1/λ1, …, 1/λn}, where {λ1, …, λn} are the eigenvalues of A (counted with algebraic multiplicity).
  2. Compute the sum of the eigenvalues of A^{-1} and express it in terms of A (e.g., as tr(A^{-1})).
  3. State any assumptions needed for these equalities to hold.

Clarifying Questions to Ask Guidance

  • Clarify the random variables, distributional assumptions, independence assumptions, and desired output.
  • Show enough derivation for the interviewer to follow the reasoning.
  • Explain how you would validate the result with simulation or sensitivity checks.

What a Strong Answer Covers Guidance

  • A correct setup with definitions, formulas, and boundary conditions.
  • A step-by-step derivation or estimation plan.
  • Interpretation of the result, including uncertainty and practical limitations.
  • Checks for assumptions, edge cases, and numerical stability.

Follow-up Questions Guidance

  • How would the result change if the assumptions were relaxed?
  • Can you verify the answer with a simulation?
  • What is the most likely source of estimation error?
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