## Jordan Forms, Direct Sum, Invariant, Complementary

### Direct Sum of Vector Spaces

The span of two subspaces U and V is the smallest subspace containing both. This is the set of vectors x+y such that x ∈ U and y ∈ V. Verify that this is indeed a subspace, and it must be included in any subspace containing U and V, hence it is the span of U and V.

If U and V are disjoint (except for 0), then the span is called the sum, or the direct sum, or the direct product. Every vector in the span has a unique representation as x+y, where x comes from U and y comes from V. If this fails then x1+y1 = x2+y2, hence x1-x2 = y2-y1, and U and V intersect after all. We can also claim U and V are linearly independent. If x+y = 0 then x = -y, U contains -y, U contains y, and U and V intersect. To complete the circle, assume U and V are linearly independent. If they are not disjoint, if they share a common vector x, then we could write x + -x = 0, which contradicts independence.

In summary, the span is a direct sum (i.e. every vector in the span has a unique representation), iff U and V are disjoint, iff U and V are linearly independent spaces.

some people write the direct sum using the plus operator, as in U+V. Others write it as U*V. I prefer the * notation, because the span really is a cross product of all possible vectors in both sets. Granted, the vectors are added together, as part of an abelian group, so + makes sense too; but it's still a cross product, and I lean towards U*V.

If U and V are independent, a basis for U and a basis for V combine to form a basis for U*V.

### Invariant

A subspace is invariant under a transformation if that subspace is mapped into itself. I don't particularly like this definition, because it sounds like the subspace is fixed by the transformation, and that isn't necessarily the case. If a transformation rotates space about the z axis, The z axis is fixed, and the xy plane is invariant. So the word invariant is a little confusing, but to be fair, I can't think of another adjective that would be any better, so invariant it is.

Given a linear transformation and an eigen value w, the eigen vectors with eigen value w form an invariant vector space. In fact each vector in that space remains in position; it is simply scaled by the eigen value w.

Zero is always invariant.

Show that the span and intersection of two invariant subspaces is an invariant subspace.

### Complementary

Subspaces are complementary if they are invariant and their direct sum produces the original space. Note that spaces are complementary with respect to a given transformation.

If our basis consists of a basis for U and a basis for V, and U and V are complementary, the transformation is implemented by a block diagonal matrix, where the upper left block operates on U, and the lower right block operates on V.