Gershgorin circle theorem

The Gershgorin circle theorem bounds every eigenvalue of a complex square matrix within a union of disks centered at its diagonal entries, with radii given by sums of off-diagonal magnitudes.

Eigenvalue: A scalar λ for which a nonzero vector v satisfies Av = λv. The theorem locates these values by comparing a largest eigenvector coordinate with its matrix row.
Nonsingular matrix: A square matrix that has an inverse, equivalently one with nonzero determinant. If zero lies outside every Gershgorin disk, the matrix must be nonsingular.
Complex number: A number expressible as a real part plus an imaginary part. Disk centers and eigenvalues may both be complex, so the theorem's geometry lives in the complex plane.
Semyon Gershgorin: A Soviet mathematician known for the eigenvalue localization theorem that bears his name. He published the disk-based eigenvalue bound in 1931.
Eigenvector: A nonzero vector whose direction is preserved by a linear transformation, changing only by a scalar factor. Choosing its largest-magnitude coordinate yields the row inequality behind the disk bound.
Iterative methods for linear systems: Algorithms that approximate solutions of linear systems through repeated updates. Diagonal-dominance conditions linked to the disks help establish convergence of common iterations.
Complex plane: A geometric plane whose points represent complex numbers by real and imaginary coordinates. Each row of a matrix defines a disk in this plane.
Matrix analysis: The study of matrices and their algebraic, geometric, and analytic properties. Gershgorin's result became a basic eigenvalue localization tool within this field.
Diagonal dominance: A matrix property in which each diagonal entry's magnitude is compared with the sum of other magnitudes in its row or column. Strict row diagonal dominance places all Gershgorin disks away from zero, excluding zero eigenvalues.
Jacobi method: An iterative method that updates each component of a linear-system solution using the previous iterate's other components. Its convergence can be guaranteed by strict row diagonal dominance, which the disks characterize spectrally.

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Sylvester's criterionCompared with: It estimates eigenvalue locations rather than testing leading principal determinants.