Page 350 - A Course in Linear Algebra with Applications

334 Chapter Nine: Advanced Topics

/
Since (VJ,VJ) = a^-, this becomes
n n
]
/(u,v) = P ^ 6 a c ,
i
i i
i
i=i j=i
from which we obtain the fundamental equation
r
/(u,v) = ([u] s ) A[v] e .
Thus the bilinear form / is represented with respect to the
basis B by the n x n matrix A whose (i,j) entry is / ( V J , V J ) .
The values of / can be computed using the above rule. In
n
particular, if / is a bilinear form on F and the standard
T
basis of F n is used, then f(X, Y) = X AY.
Conversely, if we start with a matrix A and define / by
T
means of the equation (u, v) = ([u]e) A[v]B, then it is easy
/
to verify that / is a bilinear form on V and that the matrix
representing / with respect to the basis B is A.
Now suppose we decide to use another ordered basis B':
what will be the effect on the matrix A? Let S be the invertible
matrix which describes the change of basis B' —• B. Thus
[u] B = 5[U]B', according to 6.2.4. Therefore
T
T
T
/(u,v) = (S[u} BI) A(S[v] B/) = ([u} B,) (S AS)[v] B,,
T
which shows that the matrix S AS represents / with respect
to the basis B '.
At this point we recognize that a new relation between
matrices has arisen: a matrix B is said to be congruent to a
matrix A if there is an invertible matrix S such that

T
B = S AS.

While there is an analogy between congruence and similarity
of matrices, in general similar matrices need not be congruent,
nor congruent matrices similar.