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detdt

Detdt is a term used informally to denote the derivative with respect to a parameter t of the determinant of a matrix-valued function A(t). In formal math, this quantity is written as d/dt det(A(t)) or det(A(t))′. The shorthand detdt appears in notes or discussions where the emphasis is on how the determinant changes as t varies.

The standard way to compute detdt relies on Jacobi’s formula. If A(t) is differentiable and invertible for

Example: let A(t) = I + tB for a constant matrix B. Then det(A(t))′ = det(I + tB) tr((I + tB)^{-1}

Detdt is used in areas such as sensitivity analysis, perturbation theory, and differential geometry, where understanding

t
in
a
interval,
then
det(A(t))′
=
det(A(t))
tr(A(t)^{-1}
A′(t)),
where
A′(t)
is
the
matrix
of
derivatives
of
A
with
respect
to
t.
More
generally,
without
assuming
invertibility,
det(A(t))′
=
tr(adj(A(t))
A′(t)),
where
adj(A)
is
the
adjugate
(classical
adjoint)
of
A.
These
expressions
connect
the
rate
of
change
of
the
determinant
to
the
rate
of
change
of
the
matrix
itself.
B).
At
t
=
0,
det(A(0))
=
1
and
det(A(t))′|_{t=0}
=
tr(B).
how
volumes
and
oriented
volumes
(represented
by
determinants)
change
under
deformations
is
important.
In
formal
writing,
the
same
concept
is
conveyed
with
explicit
derivative
notation
rather
than
the
shorthand
detdt.