This post will talk about methods to access different types of matrix elements (diagonals, columns, rows, etc.) when a matrix is expressed as a continguous 1D array.
Recently, I was working on implementing a matrix inversion routine using the Gauss-Jordan elimination technique in C++. This was part of the NMatrix ruby gem, and because of the limitations imposed by trying to interface a dynamic language like Ruby with C++, the elements of the NMatrix object had to expressed as a 1D contiguous C++ array for computation of the inverse.
The in-place Gauss-Jordan matrix inversion technique uses many matrix elements in every pass. Lets see some simple equations that can be used for accessing different types of elements in a matrix in a loop.
Lets say we have a square matrix A with shape M. If k is iterator we are using for going over each diagonal element of the matrix, then the equation will be something like .
A for loop using the equation should look like this:
1 2 3 4 5
To iterate over each element in a given row of a matrix, use . Here
row is the fixed row and
col goes from 0 to M-1.
To iterate over each element in a given column of a matrix, use . Here
col is the fixed column and
row goes from 0 to M-1.
In general the equation will yield a matrix element with row index
row and column index