Understanding Upper Triangular, Lower, and Diagonal Matrices

[JasonRox]

I know what an Augmented Matrix is, but what is Upper Triangular Matrix(or Lower) and a Diagonal Matrix?

This would help.

 

[Hurkyl]

A diagonal matrix is a matrix whose nonzero entries lie on the diagonal.

<br /> \left(<br /> \begin{array}{ccc}<br /> 1&amp;0&amp;0 \\<br /> 0&amp;5&amp;0 \\<br /> 0&amp;0&amp;0<br /> \end{array}<br /> \right)<br />

An upper triangular matrix is one whose nonzero entries all lie on or above the diagonal.

<br /> \left(<br /> \begin{array}{ccc}<br /> 1&amp;14&amp;0 \\<br /> 0&amp;5&amp;-3 \\<br /> 0&amp;0&amp;0<br /> \end{array}<br /> \right)<br />

 

[JasonRox]

Thanks.

Now, I understand why they keep flipping them around. Seems useless to do that, but I guess that's how its done.

 

[mathwonk]

matt grime might tell us the geometry, or other special interest of these classes of matrices. i.e. words like "cartan subgroup" or "parabolic subgroup" and so on come to mind...

strictly upper triangular matrices are also interesting because they are "nilpotent" hence counterexamples to diagonalizability.