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Lecture 31
Andrei Antonenko
April 28, 2003
1 Symmetric matrices
As we saw before, sometimes it happens that the matrix is not diagonalizable over
R
.. It may
happen, for example, when some roots of the characteristic polynomial are complex. In this
lecture we will consider the properties of symmetric matrices.
Theorem 1.1.
Let
A
be a real symmetric matrix. Then each root
λ
of its characteristic
polynomial is real.
Moreover, for symmetric matrices, eigenvectors, corresponding to diﬀerent eigenvalues are
not only linearly independent, but even orthogonal:
Theorem 1.2.
Let
A
be a real symmetric matrix. Let
λ
1
and
λ
2
be diﬀerent eigenvalues of
A
,
and
e
1
and
e
2
are corresponding eigenvectors. Then
e
1
and
e
2
are orthogonal, i.e.
h
e
1
,e
2
i
= 0
.
So, these 2 theorems give us the following main result about symmetric matrices:
Theorem 1.3.
Let
A
be a real symmetric matrix. Then there exists a matrix
C
such that
D
=
C

1
AC
is diagonal. Moreover, columns of a matrix
C
are orthogonal.
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This note was uploaded on 05/28/2011 for the course AMS 2010 taught by Professor Andant during the Spring '03 term at SUNY Stony Brook.
 Spring '03
 Andant

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