Derive eigenvalues and sum for inverse matrix

Q: Derive eigenvalues and sum for inverse matrix

This question evaluates understanding of matrix spectral properties, specifically the relationship between the eigenvalues of an invertible matrix and those of its inverse, plus familiarity with matrix invariants like the trace.

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Question

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

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

Derive eigenvalues and sum for inverse matrix

Eigenvalues of an Inverse and Their Sum

Context

Tasks

Solution

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Derive eigenvalues and sum for inverse matrix

Overview

Eigenvalues of an Inverse and Their Sum

Context

Tasks

Solution

Comments (0)