Properties of a Triangular Matrix

Brief Summary of the details of Triangular Matrices, focusing upon the Theorem of its applications.

1. PROPERTIES OF A TRIANGULAR MATRIX

Introduction

A Square Matrix is Upper Triangular (otherwise just known as
Triangular) if all entries below the diagonal of aij have the value of
Zero.

This is more formally written as:

Calculating with Triangular Matrices

Given two n by n Triangular Matrices of A = [aij] and B = [bij], then:

A + B = ((a11 + b11), …, (ann + bnn))

AB = ((a11b11), …, (annbnn))

kA = (k(a11), …, k(ann)), where k is a constant.


Functions with Triangular Matrices

Given a Triangular Matrix of A = [aij] and any polynomial function
of f(x), the result of f(A) is Triangular with the following
properties:

If i < j: aij remains the same
If i = j: aij becomes f(aij)


Inverting a Triangular Matrix

An n by n Triangular Matrix is only Invertible if:

For all i = j: aij ≠ 0

It can be asserted that, given the inverse is present, the inverse
must, also, be a Triangular Matrix.
aij = 0 if i > j