Editing Change of basis (section)

== Bilinear forms ==
A ''[[bilinear form]]'' on a vector space ''V'' over a [[field (mathematics)|field]] {{mvar|F}} is a function {{math|''V'' × ''V'' → F}} which is [[linear map|linear]] in both arguments. That is, {{math|''B'' : ''V'' × ''V'' → F}} is bilinear if the maps
<math>v \mapsto B(v, w)</math>
and <math>v \mapsto B(w, v)</math>
are linear for every fixed <math>w\in V.</math>

The matrix {{math|'''B'''}} of a bilinear form {{mvar|B}} on a basis <math>(v_1, \ldots, v_n) </math> (the "old" basis in what follows) is the matrix whose entry of the {{mvar|i}}th row and {{mvar|j}}th column is <math>B(v_i, v_j)</math>. It follows that if {{math|'''v'''}} and {{math|'''w'''}} are the column vectors of the coordinates of two vectors {{mvar|v}} and {{mvar|w}}, one has
:<math>B(v, w)=\mathbf v^{\mathsf T}\mathbf B\mathbf w,</math>
where <math>\mathbf v^{\mathsf T}</math> denotes the [[transpose]] of the matrix {{math|'''v'''}}.

If {{mvar|P}} is a change of basis matrix, then a straightforward computation shows that the matrix of the bilinear form on the new basis is
:<math>P^{\mathsf T}\mathbf B P.</math>

A [[symmetric bilinear form]] is a bilinear form {{mvar|B}} such that <math>B(v,w)=B(w,v)</math> for every {{mvar|v}} and {{mvar|w}} in {{mvar|V}}. It follows that the matrix of {{mvar|B}} on any basis is [[symmetric matrix|symmetric]]. This implies that the property of being a symmetric matrix must be kept by the above change-of-base formula. One can also check this by noting that the transpose of a matrix product is the product of the transposes computed in the reverse order. In particular,
:<math>(P^{\mathsf T}\mathbf B P)^{\mathsf T} = P^{\mathsf T}\mathbf B^{\mathsf T} P,</math>
and the two members of this equation equal <math>P^{\mathsf T} \mathbf B P</math> if the matrix {{math|'''B'''}} is symmetric.

If the [[characteristic (algebra)|characteristic]] of the ground field {{mvar|F}} is not two, then for every symmetric bilinear form there is a basis for which the matrix is [[diagonal matrix|diagonal]]. Moreover, the resulting nonzero entries on the diagonal are defined up to the multiplication by a square. So, if the ground field is the field <math>\mathbb R</math> of the [[real number]]s, these nonzero entries can be chosen to be either {{math|1}} or {{math|–1}}. [[Sylvester's law of inertia]] is a theorem that asserts that the numbers of {{math|1}} and of {{math|–1}} depends only on the bilinear form, and not of the change of basis.

Symmetric bilinear forms over the reals are often encountered in [[geometry]] and [[physics]], typically in the study of [[quadric]]s and of the [[inertia]] of a [[rigid body]]. In these cases, [[orthonormal bases]] are specially useful; this means that one generally prefer to restrict changes of basis to those that have an [[orthogonal matrix|orthogonal]] change-of-base matrix, that is, a matrix such that <math>P^{\mathsf T}=P^{-1}.</math> Such matrices have the fundamental property that the change-of-base formula is the same for a symmetric bilinear form and the endomorphism that is represented by the same symmetric matrix. The [[Spectral theorem]] asserts that, given such a symmetric matrix, there is an orthogonal change of basis such that the resulting matrix (of both the bilinear form and the endomorphism) is a diagonal matrix with the [[eigenvalues]] of the initial matrix on the diagonal. It follows that, over the reals, if the matrix of an endomorphism is symmetric, then it is [[diagonalizable matrix|diagonalizable]].