Show That a1 Is Again Column Diagonally Dominant

Bracket of matrices

In mathematics, a square matrix is said to be diagonally dominant if, for every row of the matrix, the magnitude of the diagonal entry in a row is larger than or equal to the sum of the magnitudes of all the other (not-diagonal) entries in that row. More precisely, the matrix A is diagonally ascendant if

${\displaystyle |a_{ii}|\geq \sum _{j\neq i}|a_{ij}|\quad {\text{for all }}i\,}$

where a _ij denotes the entry in the ith row and jth column.

Notation that this definition uses a weak inequality, and is therefore sometimes called weak diagonal authority. If a strict inequality (>) is used, this is called strict diagonal dominance. The unqualified term diagonal dominance tin mean both strict and weak diagonal dominance, depending on the context.^[i]

Variations [edit]

The definition in the offset paragraph sums entries across each row. It is therefore sometimes chosen row diagonal authority. If i changes the definition to sum down each column, this is called column diagonal dominance.

Whatever strictly diagonally dominant matrix is trivially a weakly chained diagonally ascendant 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 ascendant, but strictly diagonally ascendant in at least one row.

Examples [edit]

The matrix

${\displaystyle A={\begin{bmatrix}3&-2&1\\i&-3&2\\-ane&ii&four\stop{bmatrix}}}$

is diagonally dominant because

${\displaystyle |a_{xi}|\geq |a_{12}|+|a_{13}|}$ since ${\displaystyle |+three|\geq |-2|+|+1|}$

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

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

The matrix

${\displaystyle B={\begin{bmatrix}-2&2&1\\one&three&ii\\1&-ii&0\cease{bmatrix}}}$

is not diagonally dominant because

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

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

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

That is, the start and third rows fail to satisfy the diagonal potency condition.

The matrix

${\displaystyle C={\begin{bmatrix}-iv&2&1\\1&vi&ii\\ane&-two&5\end{bmatrix}}}$

is strictly diagonally dominant considering

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

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

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

Applications and properties [edit]

A strictly diagonally dominant matrix (or an irreducibly diagonally dominant matrix^[two]) is not-singular. This result is known as the Levy–Desplanques theorem.^[iii]

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

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

contradicting the hypothesis.

A Hermitian diagonally ascendant 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 perchance for ${\displaystyle A}$ . This shows that ${\displaystyle \mathrm {det} (A)\geq 0}$ . Applying this argument to the master 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

${\displaystyle {\begin{pmatrix}-2&ii&ane\end{pmatrix}}{\begin{pmatrix}1&ane&0\\1&1&0\\i&0&one\cease{pmatrix}}{\begin{pmatrix}-ii\\2\\1\terminate{pmatrix}}<0.}$

Notwithstanding, the real parts of its eigenvalues remain non-negative past 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 ascendant 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 ascendant.

A slight variation on the idea of diagonal say-so is used to show 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 authority 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 higher up sense.)

Notes [edit]

1. ^ For instance, Horn and Johnson (1985, p. 349) utilize information technology to mean weak diagonal dominance.
2. ^ Horn and Johnson, Thm vi.2.27.
3. ^ Horn and Johnson, Thm 6.1.ten. This result has been independently rediscovered dozens of times. A few notable ones are Lévy (1881), Desplanques (1886), Minkowski (1900), Hadamard (1903), Schur, Markov (1908), Rohrbach (1931), Gershgorin (1931), Artin (1932), Ostrowski (1937), and Furtwängler (1936). For a history of this "recurring theorem" see: Taussky, Olga (1949). "A recurring theorem on determinants". American Mathematical Monthly. The American Mathematical Monthly, Vol. 56, No. 10. 56 (x): 672–676. doi:x.2307/2305561. JSTOR 2305561. Another useful history is in: Schneider, Hans (1977). "Olga Taussky-Todd'south influence on matrix theory and matrix theorists". Linear and Multilinear Algebra. five (3): 197–224. doi:10.1080/03081087708817197.
4. ^ K.H. Ko and Fifty. Smolinski (1991). "A combinatorial matrix in 3-manifold theory". Pacific J. Math. 149: 319–336.

References [edit]

• Golub, Gene H.; Van Loan, Charles F. (1996). Matrix Computations. ISBN0-8018-5414-8.
• Horn, Roger A.; Johnson, Charles R. (1985). Matrix Analysis (Paperback ed.). Cambridge University Printing. ISBN0-521-38632-2.

External links [edit]

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

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Source: https://en.wikipedia.org/wiki/Diagonally_dominant_matrix

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