Cauchy Eigenvalue Interlacing

Worked Example

Consider the 3x3 symmetric "Path" matrix:

[ 2  1  0 ] [ 1  2  1 ] [ 0  1  2 ]

1. Calculate the eigenvalues of the original 3x3 matrix. The characteristic equation is det(A - λI) = (2-λ)³ - 2(2-λ) = 0. The eigenvalues are λ₁ = 2 - √2 ≈ 0.585786, λ₂ = 2, λ₃ = 2 + √2 ≈ 3.41421.
2. Delete the first row (index 0) and the first column to get the 2x2 principal submatrix:

   [ 2  1 ] [ 1  2 ]

3. Calculate the eigenvalues of this 2x2 submatrix. The characteristic equation is det(M - μI) = (2-μ)² - 1 = 0. The eigenvalues are μ₁ = 1, μ₂ = 3.
4. Compare the two sets of eigenvalues. We see that they interlace: λ₁ (0.585786) ≤ μ₁ (1) ≤ λ₂ (2) ≤ μ₂ (3) ≤ λ₃ (3.41421).

Experiment

Try it out in the interactive module:

1. Open the Cauchy Eigenvalue Interlacing module.
2. Set the Matrix control to "Path".
3. Change the Deleted index to 1. Notice how the minor eigenvalues change to 2 and 2, and the smallest interlacing slack becomes exactly 0. The interlacing inequality is still satisfied.

Limitations

This theorem specifically applies to principal submatrices of Hermitian or real symmetric matrices. If we delete a row and a different column, the resulting submatrix might not even be symmetric, and its eigenvalues will generally not interlace the original spectrum. Similarly, non-symmetric matrices do not have guaranteed real eigenvalues or interlacing properties. Numerical precision limits our ability to perfectly represent exact equality (slack of 0) when computing eigenvalues algebraically containing irrational numbers.

References

For a detailed proof and more information on interlacing inequalities:

• Eigenvalue Inequalities for Hermitian Matrices by Nicholas J. Higham explains Cauchy's interlacing theorem and other spectral inequalities.