2.2 Upper Triangular Matrices

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 Matrix 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]