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The square roots of 2x2 invertible matrices.

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:

Theorem (1):

If a 2x2 invertible matrix a is positive definite and has distinct eigenvalues then the square root of a is given by

[square root of A] = [b.sub.1] a + [b.sub.0] I. where [b.sub.0], [b.sub.1] [member of] E.

Proof:

Suppose on the contrary that A is positive definite, has distinct eigenvalues and [square root of A] [not equal to] [b.sub.1]A + [b.sub.0]I.

Since the matrix A has distinct eigenvalues so A is diagonalizable, also since A is positive definite and diagonalizable so A has a square root. [2]

Say [square root of A] = B where B is also a 2x2 matrix.

But [square root of A] [not equal to] [b.sub.1] A + [b.sub.0] I (by assumption), so

B [not equal to] [b.sub.1] [B.sup.2] + [b.sub.0] I.

So [b.sub.1] [B.sup.2] [not equal to] B - [b.sub.0] I.

Hence [B.sup.2] [not equal to] [1/[b.sub.1]] B - [[b.sub.0]/[b.sub.1]] I. ([b.sub.1] [not equal to] 0)

Take [1/[b.sub.1]] = [c.sub.1] and -[b.sub.0]/[b.sub.1] = [c.sub.0]. where [c.sub.0], [c.sub.t] [member of] R.

So [B.sup.2] [not equal to] [c.sub.1] B + [c.sub.0] I. Which contradicts Cayley- Hamilton theorem since for any 2x2 matrix A we have

[A.sub.r] = [b.sub.1] A + [b.sub.0] I, for r [greater than or equal to] 2. where [b.sub.0], [b.sub.1] [member of] R.

We now generalize the result above to any n[member of]N

Theorem (2):

If a 2x2 invertible matrix a is positive definite and has distinct eigenvalues then

[([square root of A]).sup.n] is of the form [[gamma].sub.1] A + [[gamma].sub.0] I, [for all] n[member of]N where [[gamma].sub.0], [[gamma].sub.1] [member of] R.

Proof:

By using mathematical induction

For n = 1 (done by theorem 1)

Assume it is true for n = p, so [([square root of A]).sup.p] = [a.sub.1] A + [a.sub.0] I, where [a.sub.0] I, [a.sub.1] [member of] R.

Now we want to show that it is true for n = p + 1

So [([square root of A]).sup.p+1] = [([square root of A]).sup.p].[square root of A] = ([a.sub.1] A + [a.sub.0] I) ([b.sub.1] A + [b.sub.0] I) by theorem (1)

[??] [([square root of A]).sup.p+1] = [a.sub.1][b.sub.1] [A.sup.2] + [a.sub.1][b.sub.0]A + [a.sub.0][b.sub.1]A + [a.sub.0] [b.sub.0] I.

But [A.sup.2] = [c.sub.1] A + [c.sub.0] I, where [c.sub.0], [c.sub.1] [member of] R. (using Cayley-Hamilton theorem).

Hence [([square root of A]).sup.p+1] = [a.sub.1][b.sub.1] ([c.sub.1]A + [c.sub.0]I) + ([a.sub.1][b.sub.0] + [a.sub.0][b.sub.1]) A + [a.sub.0][b.sub.0]I.

= ([a.sub.1][b.sub.1][c.sub.1] + [a.sub.1][b.sub.0] + [a.sub.0][b.sub.1]) A + ([a.sub.1][b.sub.1][c.sub.0] + [a.sub.0][b.sub.0]) I.

= [[alpha].sub.1]A + [[alpha].sub.0]I. where [[alpha].sub.0],[[alpha].sub.1] [member of] R.

Hence it is true for n = p + 1.

Another proof:

We know from Cayley-Hamilton theorem that [A.sup.r] = [[gamma].sub.1] A + [[gamma].sub.0] I. [for all] r [greater than or equal to] 2, for any 2x2 matrix and integer r.

We want to show that [A.sup.(n/2)] = [[alpha].sub.1] A + [[alpha].sub.0] I [for all] n [member of] N.

(1) If n = 1 (we are done by above).

(2) If n is even (nothing to do) as this is Cayley-Hamilton theorem

(3) If n > 1 and n is odd then

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Example:

Let [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], then

We have to find [square root of A] by using theorem (1)

p([lambda]) = [absolute value of A-[lambda]I]=0 [??] [lambda] = 1 or [lambda] = 3.

Since [square root of A] = [b.sub.1]A + [b.sub.0]I, then by using Cayley- Hamilton theorem we can replace A by [lambda] such that:

[square root of [lambda] = [b.sub.1][lambda] + [b.sub.0]

[??] 1 = [b.sub.1] + [b.sub.0] ... (1)

[square root of 3] = 3[b.sub.1] + [b.sub.0] ... (2)

Solving for [b.sub.0] and [b.sub.1] we get:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

References

[1] Alan Jeffrey, Linear Algebra and Ordinary Differential Equations, CRC press, Boca Raton, (2000).

[2] Bernard W. Levinger, Mathematics magazine, Vol. 53, No 4 (sep, 1980), pp. 222-224.

[3] Donald Sullivan, Mathematics magazine, Vol. 66, No 5 (Dec, 1993), pp, 314- 316.

[4] Higham, N.J, Newtons Method for the Matrix square Root, Math of Computation, 46 (1986) 537-549.

[5] Nick MacKinnon, Four routes to matrix roots, Math. Gaz. 73 (1989), 135-136.

Ihab Ahmad Abd AL-Baset AL-Tamimi (1*)

(1*) Directorate of Education, Huda Abd AL-Nabi AL-Natsheh School, Palestine, Hebron

E-mail: ihab.tamimi@live.com
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Author:Tamimi, Ihab Ahmad Abd AL-Baset AL-
Publication:International Journal of Difference Equations
Date:Jun 1, 2011
Words:935
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