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Matrix-valued Laplace transform
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Past exam of the mathematics course of the University of Cambridge
/
2023
/
ib
/
Paper 4
/
12B
/
a
/
Solution
Created
2026-09-24
Updated
2026-09-25
View more
Because the
matrix-valued Laplace transform
is taken componentwise, for every
i
,
j
we
have
(
L
{
A
B
}
(
s
)
)
ij
=
∫
0
∞
e
−
s
t
k
=
1
∑
n
A
ik
B
kj
(
t
)
d
t
=
k
=
1
∑
n
A
ik
∫
0
∞
e
−
s
t
B
kj
(
t
)
d
t
=
(
A
L
{
B
}
(
s
)
)
ij
.
(1)
The
sum
is finite and
A
is constant, so it may be moved outside the
integral
. Therefore
L
{
A
B
}
=
A
L
{
B
}
.
(2)
Total
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:
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