This is a slightly more complicated question. We know that the eigenvectors of a matrix are the columns of the matrix. So if we want to find out how many of these columns there are, we have to look at the dimension of each column.
Now, let’s look at our matrix A:
The first column has three elements (1 + i + j), which means that it has three dimensions. The second column has one element (1), so it only has one dimension. The third column has two elements (2 + 3j), so it has two dimensions. In total, our matrix has four dimensions: three for each column and one for the entire row.
The eigenspace for a given pair of numbers generally does not have those exact same dimensions as those numbers do individually; it is only guaranteed to be “close” to them in this way.
Last modified: September 19, 2022