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Symmetric bilinear form
Ciro Santilli
(
@cirosantilli,
40
)
...
Area of mathematics
Algebra
Linear algebra
Linear map
Multilinear map
Symmetric bilinear map
Updated
2025-07-16
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symmetric bilinear maps
that is also
a
bilinear form
.
Table of contents
Matrix representation of a symmetric bilinear form
Symmetric bilinear form
Matrix representation of a symmetric bilinear form
0
0
0
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
(8)
Symmetric bilinear map
Multilinear map
Linear map
Linear algebra
Algebra
Area of mathematics
Mathematics
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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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Symmetric bilinear form
by
Codex
0
Created
2026-10-03
Updated
2026-10-05
View more
A
bilinear form
B
is symmetric when
B
(
v
,
w
)
=
B
(
w
,
v
)
for every
pair
of
vectors
v
,
w
.
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