International Journal of Difference Equations.
ISSN 0973-6069 Volume 6, Number 1 (2011), pp. 61-64
© Research India Publications
http://www.ripublication.com/ijde.htm
The Square Roots of 2x2 Invertible Matrices
Ihab Ahmad Abd AL-Baset AL-Tamimi1*
1*
Directorate of Education , Huda Abd AL-Nabi AL-Natsheh School,
Palestine, Hebron
E-mail: [email protected]
Abstract
In this study we introduce a new method for finding the square root of a 2 2
invertible matrix using Cayley-Hamilton theorem , provided that the matrix
has distinct eigenvalues (diagonalizable) and it is positive definite .
Also we introduce a general form for √
where
, with its proof
in two ways, the new thing here is when is odd since when is even the
result is true by Cayley-Hamilton theorem.
If we have a square matrix such that .
, then we say that the
. where two matrices
square root of the matrix is the matrix . i.e √
and have the same order.
Let us now consider the following example:
1 2
1 2 1 2
7 10
Let
, then .
.
.
3 4
3 4 3 4
15 22
7 10
1 2
Hence √ =
=
.
15 22
3 4
We know that Cayley-Hamilton theorem can be reduced to :
=
+
+…+
+
, for
. where
are integers.
=
+
, for
2.
So if we have any 2 2 matrix , then
( where ,
).
Introduction
If we change the problem into system of equations, then we get inconsistent system of
equations. We solve this problem by using a system of two equations with two
unknowns.
The following theorem solves our problem:
62
Ihab Ahmad Abd AL-Baset AL-Tamimi
Theorem (1):
If a 2 2 invertible matrix is positive definite and has distinct eigenvalues then the
square root of is given by
+
. where
,
.
√ =
Proof:
Suppose on the contrary that
+
.
√
is positive definite , has distinct eigenvalues and
Since the matrix has distinct eigenvalues so is diagonalizable, also since
positive definite and diagonalizable so has a square root. 2
Say √ = where is also a 2 2 matrix .
But √
+
(by assumption), so
+
.
So
.
Hence
.(
0)
Take
=
So
2 2 matrix
and
=
,
. where
.
. Which contradicts Cayley-Hamilton theorem since for any
we have
, for
2 . where
,
.
We now generalize the result above to any
Theorem (2):
If a 2 2 invertible matrix
is of the form
√
is positive definite and has distinct eigenvalues then
,
where ,
.
Proof:
By using mathematical induction
For
1 (done by theorem 1)
Assume it is true for
, so √
Now we want to show that it is true for
So √
is
√
.√
(
)(
√
But
, where
Hence √
(
,
)
1.
,
.
) by theorem (1)
.
. ( using Cayley-Hamilton theorem).
(
Hence it is true for
, where
1
. where
(
)
)
,
(
.
.
) .
The Square Roots of 2
2 Invertible Matrices
63
Another proof:
We know from Cayley-Hamilton theorem that
2 2 matrix and integer .
.
2 , for any
We want to show that
.
(1) If
1 (we are done by above).
(2) If is even ( nothing to do ) as this is Cayley-Hamilton theorem
(3) If
1 and is odd then
.
. Where
,
,
,
.√ .
, , and
Example:
2 1
, then
1 2
We have to find √ by using theorem ( 1 )
|
p λ
λI| 0
λ 1 or
3.
Let
Since √
b
by λ such that:
√λ
b
, then by using Cayley-Hamilton theorem we can replace
b λ b
1 b
b
b
√3 3b
Solving for b and b we get:
√
b
√
√
√
√
…(1)
…(2)
√
and b
√
√
2 1
1 2
√3 1
√
√
√
√
√
√
1 0
0 1
√
√
1
√3
√
.
.
0
0
√
64
Ihab Ahmad Abd AL-Baset AL-Tamimi
References
[1]
[2]
[3]
[4]
[5]
Alan Jeffrey , Linear Algebra and Ordinary Differential Equations, CRC press,
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222-224.
Donald Sullivan , Mathematics magazine, Vol. 66, No 5 (Dec,1993), pp , 314316.
Higham, N.J , Newtons Method for the Matrix square Root, Math of
Computation, 46 (1986) 537-549.
Nick MacKinnon, Four routes to matrix roots, Math. Gaz. 73 (1989) , 135 –
136.