In: Advanced Math
Let A∈Rn× n be a non-symmetric matrix.
Prove that |λ1| is real, provided that |λ1|>|λ2|≥|λ3|≥...≥|λn| where λi , i= 1,...,n are the eigenvalues of A, while others can be real or not real.
Consider the characteristic equation .
The eigenvalues of the matrix
are the roots of this
characteristic equation.
Suppose that
are the eigenvalues. Suppose that
is not real;
then
for some reals
with
. Since
is a
polynomial equation with real coefficients, and since
is a root of this polynomial, its conjugate
is also a root of
this polynomial. Thus,
for
some
. Note
that
since
; however,
.
Thus, whenever a complex eigenvalue occurs then its absolute
value is attained by at least two distinct complex eigenvalues. In
particular, if the eigenvalue is such that
, then
must be
real.
To show that the other eigenvalues can be real or complex, we note that
and
are both non-symmetric; while the eigenvalues of are
,
those of
are
. Thus, the
other eigenvalues can be real or complex.