TOPIC 3. DIAGONALIZATION OF ENDOMORPHISMS
Mathematics – 2011/2012 - 21 -
LESSON 3
DIAGONALIZATION OF ENDOMORPHISMS
THE PROBLEM OF MATRIX DIAGONALIZATION
In this lesson we only consider square matrices. If A is an n x n matrix, then
for X in
n
ℜ
, AX is also a vector in
n
ℜ
. A question of considerable
importance in many applied problems is the determination of vectors X
such that X and AX are parallel.
Such questions arise in many economic applications, such as “Continuous
Growth Models”, etc.
A matrix diagonalization is the process of taking a square matrix and
converting it into a special type of matrix, the so-called diagonal matrix,
which shares the same fundamental properties of the underlying matrix.
Matrix diagonalization is equivalent to transforming the underlying system
of equations into a special set of coordinates axes in which the matrix takes
the canonical form. Diagonalizing a matrix is also equivalent to finding the
matrix’s Eigenvalues, which turn out to be precisely the entries of the
diagonalized matrix. Similarly, the eigenvectors make up the new set of
axis corresponding to the diagonal matrix.
As we already know, we can represent a linear map according to a matrix
shape. If the linear transformation maps a vector space of dimension n into
another vector space of dimension n, then the underlying matrix of a linear
map is a square matrix and it can be diagonalized.
nn
VVf →:
( )
Y AX
x y f x
=
→ =
So, diagonalizing is equivalent to changing basis in a vector space so that
the underlying matrix of the linear map is diagonal.
Consequently, we are looking for a diagonal matrix
Λ
, similar to
A
, which
represents
f
.
Remark: Two n-by-n matrices A and B are called similar if there exists an
invertible n-by-n matrix P such that
P
−1
AP = B.
So, we need to define two new topics, Eigenvalues and Eigenvectors.
