Page 354 - A Course in Linear Algebra with Applications

338 Chapter Nine: Advanced Topics

Theorem 9.3.4
Let f be a symmetric bilinear form on an n-dimensional real
vector space V. Then there is a basis BofV such that

/(u, v) = mvi H h u kv k - u k+iv k+1 UlVi

where u\,..., u n and v±,..., v n are the entries of the coordi-
nate vectors [u]g and [v]g respectively and k and I are integers
satisfying 0 < k < I < n.

Proof
Let / be represented by a matrix A with respect some basis
B' oiV. Then A is symmetric. Hence there is an orthogonal
T
matrix S such that S AS = D is diagonal, say with diagonal
entries di,... ,d n; of course these are the eigenvalues of A.
Here we can assume that d\,..., d k > 0, while d k+i,... ,di < 0
and di +i = • • • = d n = 0, by reordering the basis if necessary.
Let E be the nxn diagonal matrix whose diagonal entries are
the real numbers

1/y/di, ••-, 1/y/dk, l/y/-d k+1, ...,1/y/^dl, 1,...,1.

Then
T T T
(SE) A{SE) = E (S AS)E = EDE,

and the final product is the matrix

/ h I 0 | 0 \

B 0 I -Ii-k I 0
— I — I —
\ 0 I 0 1 0 /

Now the matrix SE is invertible, so its inverse determines a
change of basis from B' to say B. Then / will be represented
/
by the matrix B with respect to the basis B. Finally, (u, v) =