As a warm-up to the subject of this blog post, consider the problem of how to classify matrices up to change of basis in both the source () and the target (). In other words, the problem is to describe the equivalence classes of the equivalence relation on matrices given by
It turns out that the equivalence class of is completely determined by its rank . To prove this we construct some bases by induction. For starters, let be a vector such that ; this is always possible unless . Next, let be a vector such that is linearly independent of ; this is always possible unless .
Continuing in this way, we construct vectors such that the vectors are linearly independent, hence a basis of the column space of . Next, we complete the and to bases of in whatever manner we like. With respect to these bases, takes a very simple form: we have if and otherwise . Hence, in these bases, is a block matrix where the top left block is an identity matrix and the other blocks are zero.
Explicitly, this means we can write as a product
where has the block form above, the columns of are the basis of we found by completing , and the columns of are the basis of we found by completing . This decomposition can be computed by row and column reduction on , where the row operations we perform give and the column operations we perform give .
Conceptually, the question we’ve asked is: what does a linear transformation between vector spaces “look like,” when we don’t restrict ourselves to picking a particular basis of or ? The answer, stated in a basis-independent form, is the following. First, we can factor as a composite
where is the image of . Next, we can find direct sum decompositions and such that is the projection of onto its first factor and is the inclusion of the first factor into . Hence every linear transformation “looks like” a composite
of a projection onto a direct summand and an inclusion of a direct summand. So the only basis-independent information contained in is the dimension of the image , or equivalently the rank of . (It’s worth considering the analogous question for functions between sets, whose answer is a bit more complicated.)
The actual problem this blog post is about is more interesting: it is to classify matrices up to orthogonal change of basis in both the source and the target. In other words, we now want to understand the equivalence classes of the equivalence relation given by
Conceptually, we’re now asking: what does a linear transformation between finite-dimensional Hilbert spaces “look like”?