1. Definition & Examples
1. is Diagonally Dominant Matrix ?
`[[-3,0,0],[0,-2,0],[0,0,-1]]`
Solution:
A Square matrix `A` is called diagonally dominant matrix, if `A_(ii)>=sum_(i!=j)^()|A_(ij)|`
and `A` is called strictly diagonally dominant matrix, if `A_(ii)>sum_(i!=j)^()|A_(ij)|`
`A=[[-3,0,0],[0,-2,0],[0,0,-1]]`
`Row 1:|a_(11)|=|-3|=3,+|a_(12)|+|a_(13)|=+|0|+|0|=0`
`3 >= 0?` yes
`Row 2:|a_(22)|=|-2|=2,+|a_(21)|+|a_(23)|=+|0|+|0|=0`
`2 >= 0?` yes
`Row 3:|a_(33)|=|-1|=1,+|a_(31)|+|a_(32)|=+|0|+|0|=0`
`1 >= 0?` yes
So, given matrix is Diagonally Dominant
2. is Diagonally Dominant Matrix ?
`[[9,8,7],[6,5,4],[3,2,1]]`
Solution:
A Square matrix `A` is called diagonally dominant matrix, if `A_(ii)>=sum_(i!=j)^()|A_(ij)|`
and `A` is called strictly diagonally dominant matrix, if `A_(ii)>sum_(i!=j)^()|A_(ij)|`
`A=[[9,8,7],[6,5,4],[3,2,1]]`
`Row 1:|a_(11)|=|9|=9,+|a_(12)|+|a_(13)|=+|8|+|7|=15`
`9 >= 15?` no
`Row 2:|a_(22)|=|5|=5,+|a_(21)|+|a_(23)|=+|6|+|4|=10`
`5 >= 10?` no
`Row 3:|a_(33)|=|1|=1,+|a_(31)|+|a_(32)|=+|3|+|2|=5`
`1 >= 5?` no
So, given matrix is not Diagonally Dominant
This material is intended as a summary. Use your textbook for detail explanation.
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