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christopher-gratton
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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