# 对角优势矩阵

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

## 例子

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

${\displaystyle |a_{11}|\geq |a_{12}|+|a_{13}|}$   因為  ${\displaystyle |3|\geq |-2|+|1|}$
${\displaystyle |a_{22}|\geq |a_{21}|+|a_{23}|}$   因為  ${\displaystyle |-3|\geq |1|+|2|}$
${\displaystyle |a_{33}|\geq |a_{31}|+|a_{32}|}$   因為  ${\displaystyle |4|\geq |-1|+|2|}$.

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

${\displaystyle |b_{11}|<|b_{12}|+|b_{13}|}$   因為  ${\displaystyle |-2|<|2|+|1|}$
${\displaystyle |b_{22}|\geq |b_{21}|+|b_{23}|}$   因為  ${\displaystyle |3|\geq |1|+|2|}$
${\displaystyle |b_{33}|<|b_{31}|+|b_{32}|}$   因為  ${\displaystyle |0|<|1|+|-2|}$.

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

${\displaystyle |c_{11}|\geq |c_{12}|+|c_{13}|}$   因為  ${\displaystyle |-4|>|2|+|1|}$
${\displaystyle |c_{22}|\geq |c_{21}|+|c_{23}|}$   因為  ${\displaystyle |6|>|1|+|2|}$
${\displaystyle |c_{33}|\geq |c_{31}|+|c_{32}|}$   因為  ${\displaystyle |5|>|1|+|-2|}$.

## 參考資料

1. ^ For instance, Horn and Johnson (1985, p. 349) use it to mean weak diagonal dominance.
2. ^ Horn and Johnson, Thm 6.2.27.
3. ^ Horn and Johnson, Thm 6.1.10. 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. A recurring theorem on determinants. American Mathematical Monthly (The American Mathematical Monthly, Vol. 56, No. 10). 1949, 56 (10): 672–676. JSTOR 2305561. doi:10.2307/2305561. Another useful history is in: Schneider, Hans. Olga Taussky-Todd's influence on matrix theory and matrix theorists. Linear and Multilinear Algebra. 1977, 5 (3): 197–224. doi:10.1080/03081087708817197.
• Gene H. Golub & Charles F. Van Loan. Matrix Computations, 1996. ISBN 0-8018-5414-8
• Roger A. Horn & Charles R. Johnson. Matrix Analysis, Cambridge University Press, 1985. ISBN 0-521-38632-2 (paperback).