Mathematics

Orthogonal Decompositions

In this tutorial we learn that over an arbitrary field, every finite-dimensional bilinear space has an orthogonal decomposition into subspaces of dimension at most two. Over a field of characteristic not two, every finite-dimensional bilinear space possesses an orthogonal basis.

To further investigate bilinear forms, we want to take a closer look at orthogonal sums of bilinear modules. For the definition of an orthogonal sum, see I.1.4. Theorem I.1.9 explained when an orthogonal sum is non-degenerate or regular. Following I.2, we add that for finitely generated free bilinear modules $M_i$, the rank \[rk(M_1 \oplus M_2 \oplus ... \oplus M_n) = \sum rk(M_i)\] and \[det(M_1 \oplus M_2 \oplus ... \oplus M_n) = \prod det(M_i).\]

In theorem I.1.10 we saw an example of an orthogonal decomposition over fields. A similar result is true (and important) for bilinear modules:

Lemma I.3.1

Let $(M,b)$ be a symmetric bilinear module and $N \subseteq M$. If $b_{\mid N}$ is regular, then M is isomorphic to the orthogonal sum $N \oplus N^\perp$.

Theorem I.3.2

Let $(M,b)$ be a symmetric bilinear module and $X$ a free module spanned by linear independent elements $x_1, x_2, ..., x_k$ (see remark after definition I.2.2), then $M \cong X \oplus X^\perp$.

As an application of this theorem, we immediately get the following

Corollary I.3.3

For a finitely generated symmetric bilinear module $(M,b)$ there exists an orthogonal decomposition of the following form: $$M \cong \langle u_1 \rangle \oplus \langle u_2 \rangle \oplus ... \oplus \langle u_k \rangle \oplus N,$$ where the $u_i$ are units and $b(n,n)$ is not a unit for every $n \in N$.

Investigating this module $N$ a bit more precisely, we notice that over a field $N$ must be symplectic, thus over a field of characteristic not equal to two, the bilinear form restricted to $N$ must be zero (symmetric and symplectic). In a general setting, we get the following

Corollary I.3.4

Over a local ring of characteristic not two every symmetric bilinear space possesses an orthogonal basis.

Remark

The proof is relatively easy if we remember that every finitely generated projective module over a local ring is free.

As a last example or theorem in this rather short "chapter", we give an orthogonal decomposition for vector spaces over fields (basically the previous corollary with different words):

Corollary I.3.5

Over an arbitrary field, every finite-dimensional bilinear space has an orthogonal decomposition into subspaces of dimension at most two. Over a field of characteristic not two, every finite-dimensional bilinear space possesses an orthogonal basis.