# orthogonal direct sum

Let $({V}_{1},{B}_{1})$ and $({V}_{2},{B}_{2})$ be two vector spaces^{}, each equipped with a symmetric bilinear form^{}. Form the direct sum^{} of the two vector spaces $V:={V}_{1}\oplus {V}_{2}$. Next define a symmetric bilinear form $B$ on $V$ by

$$B(({u}_{1},{u}_{2}),({v}_{1},{v}_{2})):={B}_{1}({u}_{1},{v}_{1})+{B}_{2}({u}_{2},{v}_{2}),$$ |

where ${u}_{1},{v}_{1}\in {V}_{1}$ and ${u}_{2},{v}_{2}\in {V}_{2}$. Since $B(({u}_{1},0),({u}_{2},0))={B}_{1}({u}_{1},{u}_{2})$, we see that $B={B}_{1}$ when the domain of $B$ is restricted to ${V}_{1}$. Therefore, ${V}_{1}$ can be viewed as a subspace^{} of $V$ with respect to $B$. The same holds for ${V}_{2}$.

Now suppose $(u,0)\in {V}_{1}$ and $(0,v)\in {V}_{2}$ are two arbitrary vectors. Then $B((u,0),(0,v))={B}_{1}(u,0)+{B}_{2}(0,v)=0+0=0$. In other words, ${V}_{1}$ and ${V}_{2}$ are “orthogonal^{}” to one another with respect to $B$.

From the above discussion, we say that $(V,B)$ is the *orthogonal direct sum* of $({V}_{1},{B}_{1})$ and $({V}_{2},{B}_{2})$. Clearly the above construction is unique (up to linear isomorphisms respecting the bilinear forms^{}). As vectors from ${V}_{1}$ and ${V}_{2}$ can be seen as being “perpendicular^{}” to each other, we appropriately write $V$ as

$${V}_{1}\perp {V}_{2}.$$ |

Orthogonal Direct Sums of Quadratic Spaces. Since a symmetric^{} biliner form induces a quadratic form^{} (on the same space), we can speak of orthogonal direct sums of quadratic spaces. If $({V}_{1},{Q}_{1})$ and $({V}_{2},{Q}_{2})$ are two quadratic spaces, then the orthogonal direct sum of ${V}_{1}$ and ${V}_{2}$ is the direct sum of ${V}_{1}$ and ${V}_{2}$ with the corresponding quadratic form defined by

$$Q((u,v)):={Q}_{1}(u)+{Q}_{2}(v).$$ |

It may be shown that any $n$-dimensional quadratic space $(V,Q)$ is an orthogonal direct sum of $n$ one-dimensional quadratic subspaces. The quadratic form associated with a one-dimensional quadratic space is nothing more than $a{x}^{2}$ (the form is uniquely determined by the single coefficient $a$), and the space associated with this form is generally written as $\u27e8a\u27e9$. A finite dimensional quadratic space $V$ is commonly written as

$$\u27e8{a}_{1}\u27e9\perp \mathrm{\cdots}\perp \u27e8{a}_{n}\u27e9,\text{or simply}\u27e8{a}_{1},\mathrm{\dots},{a}_{n}\u27e9,$$ |

where $n$ is the dimension^{} of $V$.

Remark. The orthogonal direct sum may also be defined for vector spaces associated with bilinear forms that are alternating (http://planetmath.org/AlternatingForm), skew symmetric or Hermitian. The construction is similar^{} to the one discussed above.

Title | orthogonal direct sum |
---|---|

Canonical name | OrthogonalDirectSum |

Date of creation | 2013-03-22 15:42:02 |

Last modified on | 2013-03-22 15:42:02 |

Owner | CWoo (3771) |

Last modified by | CWoo (3771) |

Numerical id | 9 |

Author | CWoo (3771) |

Entry type | Definition |

Classification | msc 15A63 |

Synonym | orthogonal sum |