Implications of diagonally dominant matrices

In the previous blogs (Part 1, Part 2, Part 3, Part 4), we clarified the difference and similarities between diagonally dominant matrices, weakly diagonal dominant matrices, strongly diagonally dominant matrices, and irreducibly diagonally dominant matrices.  In this blog, we enumerate what implications these classifications have.

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If a square matrix is strictly diagonally dominant

• then the matrix is non-singular [1].
• then if the matrix is symmetric with non-negative diagonal entries, the matrix is positive semi-definite [1].
• then if the matrix is the coefficient matrix for a set of simultaneous linear equations, the iterative Gauss-Seidel numerical method will always converge [2].
• then if the matrix is the coefficient matrix for a set of simultaneous linear equations, the iterative Jordan numerical method will always converge [2].
• then if the diagonal entries of the matrix are positive, the real parts of the matrix eigenvalues are positive [1].
• then if the diagonal entries of the matrix are negative, the real parts of the matrix eigenvalues are negative [1].
• then if the matrix is column dominant, no pivoting is needed for Gaussian elimination [2].
• then if the matrix is column dominant, no pivoting is needed for LU factorization [2].

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If a square matrix is irreducible diagonally dominant

1. then if the matrix is the coefficient matrix for a set of simultaneous linear equations, the iterative Gauss-Seidel numerical method will always converge.
2. then if the matrix is the coefficient matrix for a set of simultaneous linear equations, the iterative Jordan numerical method will always converge.
3. the matrix is non-singular [2].

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If a square matrix is diagonally dominant (also called weakly diagonally dominant)

1. then if the matrix is column dominant, no pivoting is needed for Gaussian elimination [3].
2. then if the matrix is column dominant, no pivoting is needed for LU factorization [3].

References

1. Briggs, Keith. “Diagonally Dominant Matrix.” FromMathWorld–A Wolfram Web Resource, created by Eric W. Weisstein.  http://mathworld.wolfram.com/DiagonallyDominantMatrix.html

2. Diagonally Dominant Matrix, see https://en.wikipedia.org/wiki/Diagonally_dominant_matrix, Last accessed on November 4, 2016.

3. “Lecture 4: A Gaussian Elimination Example”, see http://www.cs.yale.edu/homes/spielman/BAP/lect4.pdf, last accessed on November 4, 2016.