How do you triangularise a square matrix?

Question

I'm having difficulty with triangularising square matrices. Unfortunately the lecturer ran out of time today, and so wasn't able to do any examples of how to triangularise a square matrix, and I have some work to hand in tomorrow and am very stuck! I would really appreciate it if anyone could give me an example of how to triangularise a square matrix, or if you could recommend any websites that show how you would go about this.

Thanks for letting me know about the elementary row operations. Do you know if there is any way to find an invertible matrix P such that P^(-1)AP is triangular, where A is the original matrix (assuming that it's minimal polynomial is the product of linear factors)? I'd really appreciate it if you could point me in the right direction! Thanks!

Answer 1

here's an example

we want to reduce a matrix to u/t form

1 10 -3 r1
1 10 2 r2
1 4 2 r3

1 10 -3
0 0 5
0 -6 5
r2-r1
r3-r1

1 10 -3
0 -6 5
0 0 5
interchange rows 2 and 3

upper triangular form

this is known as gaussian elimination
and is used in solving systems of equations

go to the site:

http://mathworld.wolfram.com/GaussianElimination.html

this gives a good example

you can also get help with other matrix
operations at this site,plus access to the
integrator as well

i hope that this helps

Answer 2

Just use row operations. Use the first row to eliminate (make zero) all the other elements in the first column. Then use the second row to zero out all column elements below it. Continue and you'll end up with an upper triangular matrix.