(i) Explain why the eigenvalues of A must lie in its refined...

(i) Explain why the eigenvalues of $A$ must lie in its refined Gershgorin domain $D_A^*=D_{A^T} \cap D_A$. (ii) Find the refined Gershgorin domains for each of the matrices in Exercise 8.2 .53 and confirm the result in part (i).

Key Concepts

Matrix Transposition and Its Role in Eigenvalue Estimates

Considering both a matrix and its transpose in eigenvalue estimation helps in capturing complementary information about the matrix’s structure. While the original Gershgorin theorem applies to the rows of the matrix, applying it to the transpose allows one to consider the columns. The intersection of these two resulting sets typically yields a tighter inclusion region for the eigenvalues.

Eigenvalue Localization

Eigenvalue localization refers to various methods and results used to bound or approximate the regions in the complex plane where the eigenvalues of a matrix are found. The use of Gershgorin discs, and in particular the refined Gershgorin domain, are key techniques for localizing eigenvalues without needing to compute them explicitly.

Refined Gershgorin Domain

The refined Gershgorin domain is obtained by taking the intersection of the traditional Gershgorin domains of a matrix and its transpose. This refinement takes into account both the row sums and the column sums of off-diagonal entries, often leading to a smaller and more accurate region in which all the eigenvalues of the matrix must lie.

Gershgorin Circle Theorem

This theorem provides a way to estimate where the eigenvalues of a square matrix lie by constructing disks (or circles) in the complex plane. Each disk is centered at a diagonal entry of the matrix and has a radius equal to the sum of the absolute values of the non-diagonal entries in the corresponding row. The theorem guarantees that all eigenvalues of the matrix lie within the union of these disks.

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