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Diagonally Dominant Matrix


A square matrix A is called diagonally dominant if |A_(ii)|>=sum_(j!=i)|A_(ij)| for all i. A is called strictly diagonally dominant if |A_(ii)|>sum_(j!=i)|A_(ij)| for all i.

A strictly diagonally dominant matrix is nonsingular. A symmetric diagonally dominant real matrix with nonnegative diagonal entries is positive semidefinite.

If a matrix is strictly diagonally dominant and all its diagonal elements are positive, then the real parts of its eigenvalues are positive; if all its diagonal elements are negative, then the real parts of its eigenvalues are negative. These results follow from the Gershgorin circle theorem.


See also

Diagonal Matrix

This entry contributed by Keith Briggs

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Cite this as:

Briggs, Keith. "Diagonally Dominant Matrix." From MathWorld--A Wolfram Web Resource, created by Eric W. Weisstein. https://mathworld.wolfram.com/DiagonallyDominantMatrix.html

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