I have a matrix that looks like a discrete Laplacian. Now I can plot the entries of the eigenvector by their index on the x axis and value on the y-axis. Now increasing the size of the matrix makes the datapoints look like a function i.e. plotting eigenvector as a function. Now in the large matrix size limit, how do I show that this actually converges to the eigenfunction of the continuous operator and the eigenfunction is continuous? Is there some theorem showing this?
Example:
