Matlab supports 3 forms of indexing matrices. There is the familiar range indexing that selects a submatrix within a larger matrix:
A = randn(10); A(1:2,1:4) %a 2x4 matrix extracted from the top of A
This is also called sub-indexing. There is also logical indexing:
A = randn(10); A(A>1.5) = Inf; %Transform values greater than 1.5 to Inf.
These two methods are fairly expressive, yet there are a certain number of operations which are awkward or impossible to perform with these. Let’s say that we have a matrix A and we want to set its off diagonal (not the regular \ one, the one shaped like /) to all ones. You might think this requires the use of a loop or some special function within Matlab. Yet this operation is easily done in a single step with linear indexing.
Background: a Matlab matrix is internally stored as a flag defining its shape, followed by sequential numeric data organized such that the first index of the matrix grows fastest and the last grows slowest. Thus, in a 2d matrix, the elements are ordered in a И pattern. If a matrix has size (10,20), the element (1,1) has linear index 1, the element (10,1) has index 10, (1,2) has index 11, and so forth.
To target the off-diagonal of a 10×10 matrix, we write down the paired indices of the off diagonal:
(10,1), (9,2), (8,3), …
And translate them into linear indices. For a 2D matrix of size MxN, the element (i,j) has linear index equal to i + (j-1)*M. So you could select the off diagonal using:
A((10:-1:1) + (0:9)*10)
A = randn(10); A(sub2ind(size(A),size(A,1):-1:1,1:size(A,2))) = 1;
This come in handy in other situations. Say that you have a vector of parameters for a model that represent spatial gains. So for a 16×16 receptive field you have represented the data as a 256×1 vector w. Say that you want to create a matrix D that computes horizonal spatial derivatives. Here’s one way to do it that involves no for loop:
inds = sub2ind([16,16],reshape(repmat((1:16),15*2,1),2,16*15),repmat([1:15;2:16],1,16)); D = zeros(15*16,16*16); D(sub2ind([15*16,16*16],ones(2,1)*(1:15*16),inds)) = repmat([1,-1]',1,15*16);
Unreadable, but will net you a mighty amount of geek points! Now for something a little more useful. Say that you want to find the location of the maximum element in a matrix. You might use something like:
A = randn(10); [rowmax,bestrows] = max(A); [~,bestcol] = max(bestrows) %The max element is at (bestrows(bestcol),bestcol)
It’s cleaner however to use the fact that the (:) operator reshapes a vector or matrix A to size (numel(A),1), obtain the max position as a linear index, and then translate to sub-indices:
A = randn(10); [~,bestpos] = max(A(:)); [bestrow,bestcol] = ind2sub(size(A),bestpos);
I should also note that this works regardless of how many dimensions A has. So for example the largest element in a 5-dimensional matrix can be determined like so:
A = randn(5,4,3,4,6); [~,bestpos] = max(A(:)); [mi,mj,mk,ml,mm] = ind2sub(size(A),bestpos);
sub2ind and ind2sub thus can make it quite a bit easier to work with multidimensional matrices.