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by
Ciro Santilli
(@cirosantilli,
32
)
Symmetric bilinear form
symmetric bilinear maps
that is also a
bilinear form
.
Table of contents
Matrix representation of a symmetric bilinear form
Matrix representation of a symmetric bilinear form
Symmetric bilinear form
Like the
matrix representation of a bilinear form
, it is a
matrix
, but now the matrix has to be a
symmetric matrix
.
We can then immediately see that the matrix is symmetric, then so is the form. We have:
B
(
x
,
y
)
=
x
T
M
y
(1)
But because
B
(
x
,
y
)
is a
scalar
, we have:
B
(
x
,
y
)
=
B
(
x
,
y
)
T
(2)
and:
B
(
x
,
y
)
=
B
(
x
,
y
)
T
=
(
x
T
M
y
)
T
=
y
T
M
T
x
=
y
T
M
T
x
=
y
T
M
x
=
B
(
y
,
x
)
(3)
Ancestors
Symmetric bilinear map
Multilinear map
Linear map
Linear algebra
Algebra
Area of mathematics
Mathematics
Index
Incoming links
Bilinear map
Hermitian form
Minkowski inner product
Minkowski inner product matrix
Positive definite symmetric bilinear form
Quadratic form
Symmetric matrix
Symplectic group
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