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by
Ciro Santilli
(
@cirosantilli,
37
)
Matrix representation of a symmetric bilinear form
...
Algebra
Linear algebra
Linear map
Multilinear map
Symmetric bilinear map
Symmetric bilinear form
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Updated
2025-05-23
+
Created
1970-01-01
See my version
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
(9)
Symmetric bilinear form
Symmetric bilinear map
Multilinear map
Linear map
Linear algebra
Algebra
Area of mathematics
Mathematics
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What happens to the definition of the orthogonal group if we choose other types of symmetric bilinear forms
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cirosantilli/matrix-representation-of-the-symmetric-bilinear-form
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