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SEEMOUS
2012 SEEMOUS
Problem 3
Problem 3
Part of
2012 SEEMOUS
Problems
(1)
property of traces
Source: SEEMOUS 2012 P3
6/9/2021
a) Prove that if
k
k
k
is an even positive integer and
A
A
A
is a real symmetric
n
×
n
n\times n
n
×
n
matrix such that
tr
(
A
k
)
k
+
1
=
tr
(
A
k
+
1
)
k
\operatorname{tr}(A^k)^{k+1}=\operatorname{tr}(A^{k+1})^k
tr
(
A
k
)
k
+
1
=
tr
(
A
k
+
1
)
k
, then
A
n
=
tr
(
A
)
A
n
−
1
.
A^n=\operatorname{tr}(A)A^{n-1}.
A
n
=
tr
(
A
)
A
n
−
1
.
b) Does the assertion from a) also hold for odd positive integers
k
k
k
?
matrix
linear algebra