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by
Ciro Santilli
(
@cirosantilli,
34
)
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
2024-11-30
Created
1970-01-01
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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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