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vestigium

Vestigium, from the Latin word meaning trace, is the term used in linear algebra for the trace of a square matrix. For an n-by-n matrix A with entries aij over a field, vestigium(A) or tr(A) is the sum of its diagonal entries: tr(A) = a11 + a22 + ... + ann.

Key properties and interpretations

The vestigium is invariant under similarity: tr(P^{-1}AP) = tr(A) for any invertible P. It is a linear

Examples and applications

For a 2×2 matrix A = [ [1, 2], [3, 4] ], the vestigium is 1 + 4 = 5. In

See also

Trace (linear algebra).

map
on
the
space
of
square
matrices:
tr(A
+
B)
=
tr(A)
+
tr(B)
and
tr(cA)
=
c
tr(A)
for
any
scalar
c.
If
A
has
eigenvalues
λ1,
λ2,
...,
λn
(counted
with
algebraic
multiplicity),
then
the
vestigium
equals
the
sum
of
the
eigenvalues:
tr(A)
=
λ1
+
λ2
+
...
+
λn.
This
connects
to
the
characteristic
polynomial
det(λI
−
A)
=
λ^n
−
(tr
A)
λ^{n−1}
+
...;
the
coefficient
of
λ^{n−1}
is
−tr(A).
broader
contexts,
the
trace
provides
a
simple
matrix
invariant
and
appears
in
identities,
representations,
and
differential
geometry,
often
used
to
relate
a
matrix
to
its
eigenstructure.