Understanding Dominant Matrices for Year 11 Students

[Zashmar]

Can anyone explain to a year 11 student what a dominant matrix is exactly?
my textbook is not making much sense, i understand basic matricies and how you times them and rearange equations.

Thank you so much(Happy)

 

[MarkFL]

Are you referring to diagonal dominance, and if so, is it row or column?

 

[Zashmar]

Yes diagonally dominant, what does it mean

 

[MarkFL]

Let's say it is row diagonal dominance.

For the $n$th row, the magnitude of the entry in the $n$th column must be greater than or equal to the sum of the magnitudes of all other entries in that row.

This must be true for all rows. Stated mathematically:

$\displaystyle |a_{ii}|\ge\sum_{j\ne1}|a_{ij}|$ for all $i$.

where $a_{ij}$ denotes the entry in the $i$th row and $j$th column.

Note that this definition uses a weak inequality, and is therefore sometimes called weak diagonal dominance. If a strict inequality (>) is used, this is called strict diagonal dominance. The unqualified term diagonal dominance can mean both strict and weak diagonal dominance, depending on the context.

 

[blue_raver22]

Sure, I would be happy to explain what a dominant matrix is to a year 11 student. A dominant matrix is a special type of matrix that has specific properties. In order to understand what a dominant matrix is, it is important to first understand what a matrix is. A matrix is a rectangular array of numbers, with rows and columns, that is used to represent data or perform calculations.

Now, a dominant matrix is a square matrix, meaning it has the same number of rows and columns, where the absolute value of the elements in each row is greater than the absolute value of the elements in the same row of any other column. In simpler terms, this means that the largest number in each row is bigger than the largest number in the same column.

Why is this important? Dominant matrices have special properties that make them useful in solving systems of equations. In fact, they can be used to find the solution to a system of equations even when other methods may not work. This is because the dominant element in each row can be used to eliminate variables in the equations, making it easier to solve for the remaining variables.

I hope this helps clarify what a dominant matrix is and why it is important. If you have any further questions, please don't hesitate to ask.