**17.2****
****Upper Triangular Matrices** (.NET, C#, CSharp, VB, Visual Basic, F#)

An *upper triangular *matrix
is a square matrix with all elements below the main diagonal equal to
zero. That is, for
. For example, this
is a 4 x 4 upper triangular matrix:

Like lower triangular matrices, upper triangular matrices often arise at an intermediate stage in solving systems of equations and inverting matrices.

**NMath**
provides upper triangular matrix classes for four datatypes: single-
and double-precision floating point numbers, and single- and double-precision
complex numbers. The classnames are **FloatUpperTriMatrix**,
**DoubleUpperTriMatrix**,
**FloatComplexUpperTriMatrix**,
and **DoubleComplexUpperTriMatrix**.

For efficiency, zero elements below the main diagonal are not stored. Instead, matrix values are stored in a vector column by column. For example, the following 5 x 5 upper triangular matrix:

is stored in a data vector as:

v = [ a00 a01 a11 a02 a12 a22 a03 a13 a23 a33 a04 a14 a24 a34 a44 ]

In general, the relationship between matrix and vector indices is:

A[i,j] = v[i + j(j+1)/2]