Weakly chained diagonally dominant matrix

Venn Diagram showing the containment of weakly chained diagonally dominant (WCDD) matrices relative to weakly diagonally dominant (WDD) and strictly diagonally dominant (SDD) matrices.

In mathematics, the weakly chained diagonally dominant matrices are a family of nonsingular matrices that include the strictly diagonally dominant matrices.

Definition

Preliminaries

We say row ${\displaystyle i}$ of a complex matrix ${\displaystyle A=(a_{ij})}$ is strictly diagonally dominant (SDD) if ${\displaystyle |a_{ii}|>\textstyle {\sum _{j\neq i}}|a_{ij}|}$. We say ${\displaystyle A}$ is SDD if all of its rows are SDD. Weakly diagonally dominant (WDD) is defined with ${\displaystyle \geq }$ instead.

The directed graph associated with an ${\displaystyle m\times m}$ complex matrix ${\displaystyle A=(a_{ij})}$ is given by the vertices ${\displaystyle \{1,\ldots ,m\}}$ and edges defined as follows: there exists an edge from ${\displaystyle i\rightarrow j}$ if and only if ${\displaystyle a_{ij}\neq 0}$.

Definition

A complex square matrix ${\displaystyle A}$ is said to be weakly chained diagonally dominant (WCDD) if

• ${\displaystyle A}$ is WDD and
• for each row ${\displaystyle i_{1}}$ that is not SDD, there exists a walk ${\displaystyle i_{1}\rightarrow i_{2}\rightarrow \cdots \rightarrow i_{k}}$ in the directed graph of ${\displaystyle A}$ ending at an SDD row ${\displaystyle i_{k}}$.

Example

The directed graph associated with the WCDD matrix in the example. The first row, which is SDD, is highlighted. Note that regardless of which node ${\displaystyle i}$ we start at, we can find a walk ${\displaystyle i\rightarrow (i-1)\rightarrow (i-2)\rightarrow \cdots \rightarrow 1}$.

The ${\displaystyle m\times m}$ matrix

${\displaystyle {\begin{pmatrix}1\\-1&1\\&-1&1\\&&\ddots &\ddots \\&&&-1&1\end{pmatrix}}}$

is WCDD.

Properties

Nonsingularity

A WCDD matrix is nonsingular. ^[1]

Proof: ^[2] Let ${\displaystyle A=(a_{ij})}$ be a WCDD matrix. Suppose there exists a nonzero ${\displaystyle x}$ in the null space of ${\displaystyle A}$. Without loss of generality, let ${\displaystyle i_{1}}$ be such that ${\displaystyle |x_{i_{1}}|=1\geq |x_{j}|}$ for all ${\displaystyle j}$. Since ${\displaystyle A}$ is WCDD, we may pick a walk ${\displaystyle i_{1}\rightarrow i_{2}\rightarrow \cdots \rightarrow i_{k}}$ ending at an SDD row ${\displaystyle i_{k}}$.

Taking moduli on both sides of

${\displaystyle -a_{i_{1}i_{1}}x_{i_{1}}=\sum _{j\neq i_{1}}a_{i_{1}j}x_{j}}$

and applying the triangle inequality yields

${\displaystyle \left|a_{i_{1}i_{1}}\right|\leq \sum _{j\neq i_{1}}\left|a_{i_{1}j}\right|\left|x_{j}\right|\leq \sum _{j\neq i_{1}}\left|a_{i_{1}j}\right|,}$

and hence row ${\displaystyle i_{1}}$ is not SDD. Moreover, since ${\displaystyle A}$ is WDD, the above chain of inequalities holds with equality so that ${\displaystyle |x_{j}|=1}$ whenever ${\displaystyle a_{i_{1}j}\neq 0}$. Therefore, ${\displaystyle |x_{i_{2}}|=1}$. Repeating this argument with ${\displaystyle i_{2}}$, ${\displaystyle i_{3}}$, etc., we find that ${\displaystyle i_{k}}$ is not SDD, a contradiction. ${\displaystyle \square }$

Recalling that an irreducible matrix is one whose associated directed graph is strongly connected, a trivial corollary of the above is that an irreducibly diagonally dominant matrix (i.e., an irreducible WDD matrix with at least one SDD row) is nonsingular. ^[3]

Relationship with nonsingular M-matrices

The following are equivalent: ^[4]

• ${\displaystyle A}$ is a nonsingular WDD M-matrix.
• ${\displaystyle A}$ is a nonsingular WDD L-matrix;
• ${\displaystyle A}$ is a WCDD L-matrix;

In fact, WCDD L-matrices were studied (by James H. Bramble and B. E. Hubbard) as early as 1964 in a journal article ^[5] in which they appear under the alternate name of matrices of positive type.

Moreover, if ${\displaystyle A}$ is an ${\displaystyle n\times n}$ WCDD L-matrix, we can bound its inverse as follows: ^[6]

${\displaystyle \left\Vert A^{-1}\right\Vert _{\infty }\leq \sum _{i}\left[a_{ii}\prod _{j=1}^{i}(1-u_{j})\right]^{-1}}$  where  ${\displaystyle u_{i}={\frac {1}{\left|a_{ii}\right|}}\sum _{j=i+1}^{n}\left|a_{ij}\right|.}$

Note that ${\displaystyle u_{n}}$ is always zero and that the right-hand side of the bound above is ${\displaystyle \infty }$ whenever one or more of the constants ${\displaystyle u_{i}}$ is one.

Tighter bounds for the inverse of a WCDD L-matrix are known. ^{[7] [8] [9] [10]}

Applications

Due to their relationship with M-matrices (see above), WCDD matrices appear often in practical applications. An example is given below.

Monotone numerical schemes

WCDD L-matrices arise naturally from monotone approximation schemes for partial differential equations.

For example, consider the one-dimensional Poisson problem

${\displaystyle u^{\prime \prime }(x)+g(x)=0}$  for  ${\displaystyle x\in (0,1)}$

with Dirichlet boundary conditions ${\displaystyle u(0)=u(1)=0}$. Letting ${\displaystyle \{0,h,2h,\ldots ,1\}}$ be a numerical grid (for some positive ${\displaystyle h}$ that divides unity), a monotone finite difference scheme for the Poisson problem takes the form of

${\displaystyle -{\frac {1}{h^{2}}}A{\vec {u}}+{\vec {g}}=0}$  where  ${\displaystyle [{\vec {g}}]_{j}=g(jh)}$

and

${\displaystyle A={\begin{pmatrix}2&-1\\-1&2&-1\\&-1&2&-1\\&&\ddots &\ddots &\ddots \\&&&-1&2&-1\\&&&&-1&2\end{pmatrix}}.}$

Note that ${\displaystyle A}$ is a WCDD L-matrix.

References

1. Shivakumar, P. N.; Chew, Kim Ho (1974). "A Sufficient Condition for Nonvanishing of Determinants" (PDF). Proceedings of the American Mathematical Society. 43 (1): 63. doi: 10.1090/S0002-9939-1974-0332820-0 . ISSN   0002-9939.
2. Azimzadeh, Parsiad; Forsyth, Peter A. (2016). "Weakly Chained Matrices, Policy Iteration, and Impulse Control". SIAM Journal on Numerical Analysis. 54 (3): 1341–1364. arXiv: 1510.03928 . doi:10.1137/15M1043431. ISSN   0036-1429.
3. Horn, Roger A.; Johnson, Charles R. (1990). "Matrix analysis". Cambridge University Press, Cambridge.
4. Azimzadeh, Parsiad (2019). "A fast and stable test to check if a weakly diagonally dominant matrix is a nonsingular M-Matrix". Mathematics of Computation. 88 (316): 783–800. arXiv: 1701.06951 . Bibcode:2017arXiv170106951A. doi:10.1090/mcom/3347.
5. Bramble, James H.; Hubbard, B. E. (1964). "On a finite difference analogue of an elliptic problem which is neither diagonally dominant nor of non-negative type". Journal of Mathematical Physics. 43: 117–132. doi:10.1002/sapm1964431117.
6. Shivakumar, P. N.; Williams, Joseph J.; Ye, Qiang; Marinov, Corneliu A. (1996). "On Two-Sided Bounds Related to Weakly Diagonally Dominant M-Matrices with Application to Digital Circuit Dynamics". SIAM Journal on Matrix Analysis and Applications. 17 (2): 298–312. doi:10.1137/S0895479894276370. ISSN   0895-4798.
7. Cheng, Guang-Hui; Huang, Ting-Zhu (2007). "An upper bound for ${\displaystyle \Vert A^{-1}\Vert _{\infty }}$ of strictly diagonally dominant M-matrices". Linear Algebra and Its Applications. 426 (2–3): 667–673. doi: 10.1016/j.laa.2007.06.001 . ISSN   0024-3795.
8. Li, Wen (2008). "The infinity norm bound for the inverse of nonsingular diagonal dominant matrices". Applied Mathematics Letters. 21 (3): 258–263. doi:10.1016/j.aml.2007.03.018. ISSN   0893-9659.
9. Wang, Ping (2009). "An upper bound for ${\displaystyle \Vert A^{-1}\Vert _{\infty }}$ of strictly diagonally dominant M-matrices". Linear Algebra and Its Applications. 431 (5–7): 511–517. doi: 10.1016/j.laa.2009.02.037 . ISSN   0024-3795.
10. Huang, Ting-Zhu; Zhu, Yan (2010). "Estimation of ${\displaystyle \Vert A^{-1}\Vert _{\infty }}$ for weakly chained diagonally dominant M-matrices". Linear Algebra and Its Applications. 432 (2–3): 670–677. doi: 10.1016/j.laa.2009.09.012 . ISSN   0024-3795.