Diagonally dominant matrix

The definition in the first paragraph sums entries across rows. It is therefore sometimes called row diagonal dominance. If one changes the definition to sum down columns, this is called column diagonal dominance.

Any strictly diagonally dominant matrix is trivially a weakly chained diagonally dominant matrix. Weakly chained diagonally dominant matrices are nonsingular and include the family of irreducibly diagonally dominant matrices. These are irreducible matrices that are weakly diagonally dominant, but strictly diagonally dominant in at least one row.

The matrix

${\displaystyle A={\begin{bmatrix}3&-2&1\\1&-3&2\\-1&2&4\end{bmatrix}}}$

is diagonally dominant because

${\displaystyle |a_{11}|\geq |a_{12}|+|a_{13}|}$   since   ${\displaystyle |+3|\geq |-2|+|+1|}$

${\displaystyle |a_{22}|\geq |a_{21}|+|a_{23}|}$   since   ${\displaystyle |-3|\geq |+1|+|+2|}$

${\displaystyle |a_{33}|\geq |a_{31}|+|a_{32}|}$   since   ${\displaystyle |+4|\geq |-1|+|+2|}$.

The matrix

${\displaystyle B={\begin{bmatrix}-2&2&1\\1&3&2\\1&-2&0\end{bmatrix}}}$

is not diagonally dominant because

${\displaystyle |b_{11}|<|b_{12}|+|b_{13}|}$   since   ${\displaystyle |-2|<|+2|+|+1|}$

${\displaystyle |b_{22}|\geq |b_{21}|+|b_{23}|}$   since   ${\displaystyle |+3|\geq |+1|+|+2|}$

${\displaystyle |b_{33}|<|b_{31}|+|b_{32}|}$   since   ${\displaystyle |+0|<|+1|+|-2|}$.

That is, the first and third rows fail to satisfy the diagonal dominance condition.

The matrix

${\displaystyle C={\begin{bmatrix}-4&2&1\\1&6&2\\1&-2&5\end{bmatrix}}}$

is strictly diagonally dominant because

${\displaystyle |c_{11}|>|c_{12}|+|c_{13}|}$   since   ${\displaystyle |-4|>|+2|+|+1|}$

${\displaystyle |c_{22}|>|c_{21}|+|c_{23}|}$   since   ${\displaystyle |+6|>|+1|+|+2|}$

${\displaystyle |c_{33}|>|c_{31}|+|c_{32}|}$   since   ${\displaystyle |+5|>|+1|+|-2|}$.

A strictly diagonally dominant matrix (or an irreducibly diagonally dominant matrix[2]) is non-singular. This result is known as the Levy–Desplanques theorem.[3]

Proof: Suppose that A is a strictly diagonally dominant matrix and ${\displaystyle v=(v_{1},\ldots ,v_{n})}$ is a nonzero vector such that ${\displaystyle Av=0}$. Let i be such that ${\displaystyle v_{i}}$ is maximum in absolute value. Then

${\displaystyle |(Av)_{i}|=\left|\sum _{j=1}^{n}A_{ij}v_{j}\right|\geq |A_{ii}||v_{i}|-\sum _{j\neq i}|A_{ij}||v_{j}|\geq |v_{i}|\left(|A_{ii}|-\sum _{j\neq i}|A_{ij}|\right)>0,}$

contradicting the hypothesis.

A Hermitian diagonally dominant matrix ${\displaystyle A}$ with real non-negative diagonal entries is positive semidefinite.

Proof: Let the diagonal matrix ${\displaystyle D}$ contain the diagonal entries of ${\displaystyle A}$. Connect ${\displaystyle A}$ and ${\displaystyle D+I}$ via a segment of matrices ${\displaystyle M(t)=(1-t)(D+I)+tA}$. This segment consists of strictly diagonally dominant (thus nonsingular) matrices, except maybe for ${\displaystyle A}$. This shows that ${\displaystyle \mathrm {det} (A)\geq 0}$. Applying this argument to the principal minors of ${\displaystyle A}$, the positive semidefiniteness follows by Sylvester's criterion.

If the symmetry requirement is eliminated, such a matrix is not necessarily positive semidefinite. For example, consider

${\displaystyle {\begin{pmatrix}-2&2&1\end{pmatrix}}{\begin{pmatrix}1&1&0\\1&1&0\\1&0&1\end{pmatrix}}{\begin{pmatrix}-2\\2\\1\end{pmatrix}}<0.}$

However, the real parts of its eigenvalues remain non-negative by the Gershgorin circle theorem.

Similarly, an Hermitian strictly diagonally dominant matrix with real positive diagonal entries is positive definite, as it equals to the sum of some Hermitian diagonally dominant matrix ${\displaystyle A}$ with real non-negative diagonal entries (which is positive semidefinite) and ${\displaystyle xI}$ for some positive real number ${\displaystyle x}$ (which is positive definite).

No (partial) pivoting is necessary for a strictly column diagonally dominant matrix when performing Gaussian elimination (LU factorization).

The Jacobi and Gauss–Seidel methods for solving a linear system converge if the matrix is strictly (or irreducibly) diagonally dominant.

Many matrices that arise in finite element methods are diagonally dominant.

A slight variation on the idea of diagonal dominance is used to prove that the pairing on diagrams without loops in the Temperley–Lieb algebra is nondegenerate.[4] For a matrix with polynomial entries, one sensible definition of diagonal dominance is if the highest power of ${\displaystyle q}$ appearing in each row appears only on the diagonal. (The evaluations of such a matrix at large values of ${\displaystyle q}$ are diagonally dominant in the above sense.)

• Golub, Gene H.; Van Loan, Charles F. (1996). Matrix Computations. ISBN 0-8018-5414-8.
• Horn, Roger A.; Johnson, Charles R. (1985). Matrix Analysis (Paperback ed.). Cambridge University Press. ISBN 0-521-38632-2.

• PlanetMath: Diagonal dominance definition
• PlanetMath: Properties of diagonally dominant matrices
• Mathworld