Exercise 1965Topology › Topology of Mn(ℝ)M1Added 4 Nov 2026

Continuity of eigenvalues (statement)

#eigenvalues#continuity#roots of polynomial

Statement

Show that the multiset of eigenvalues of a matrix depends continuously on the matrix entries.

Hint

Eigenvalues are roots of the characteristic polynomial $\chi_A(\lambda)=\det(\lambda I-A)$ , whose coefficients depend continuously on $A$ ; the roots of a monic polynomial depend continuously on its coefficients.

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Solution

Characteristic polynomial. $A\mapsto\chi_A$ is continuous (composition of $\det$ and affine maps).

Root continuity. A monic polynomial's roots (with multiplicity) depend continuously on the coefficients in the Hausdorff metric on finite multisets of $\mathbb{C}$ .

Conclusion. Eigenvalues depend continuously on the matrix.

Continuity of eigenvalues (statement)

Statement

Solution

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