Una transformación de similaridad para la matriz de rotación - ijmes

International Journal of Mathematical Engineering and Science 

ISSN : 2277-6982 Volume 1 Issue 1 

http://www.ijmes.com/ https://sites.google.com/site/ijmesjournal/ 

A SIMILARITY TRANSFORMATION FOR THE 

ROTATION MATRIX 

J. López-Bonilla 1 , A. Hernández-Galeana 2 , J. Rivera-Rebolledo, 

1 ESIME-Zacatenco, Instituto Politécnico Nacional (IPN), 

Anexo Edif.3, Col. Lindavista, 07738 México DF, 

2 Depto. de Física, Escuela Superior de Física y Matemáticas – IPN, 

Edif. 9, Zacatenco, Col. Lindavista, 07738 México DF 

jlopezb@ipn.mx 

ABSTRACT. It is shown explicitly a similarity transformation to diagonalize the 

rotation matrix. 

Keywords: Rotation matrix, Similarity transformation, Cayley-Klein parameters. 

1. Introduction 

The 3- space rotations can be performed with the matrix [1]: 

12222 2 x 2 x i 2 x 2 x x x 

 

 

2 2 

 

 

i 

2 2 2 2 

2 x 2 x 1 2 x 2 x x x 

R i 

2 2 

 

x x x x x x 

i 

 

 

 

 

where and are the Cayley-Klein parameters under the condition: 

8 

, (1) 

x x 

1 , (2) 

which implies the orthogonal character of (1): 

T 

RR I , det R 1. 

(3) 

The Euler-Olinde Rodrigues coefficients [2],[3] a1,..., a 4 defined by: 

i x 1 x i 

x 1 x 

a1 , a2 , a3 , a4 

, 

(4) 

2 2 2 2




verify: 

by virtue of (2), so that (1) adopts the form: 

2 2 2 2 

a1a2a3a4 1 , 

(5) 

 

2 2 a2 a3 a1a 2 a3a4 a1a3 a2a4 

1 2 3 4 2 

1 

2 

3 2 3 1 4 

a1a 3 a2a4 a1a 4 a2a3 

2 2 a1 a2 

 

1222 

R 2aaaa12aa2 a a a a , 

 

2212 

The rotation matrix represents [4] a rotation of the Cartesian axes through a Φ angle around the 

l , l , l , such that [3]: 

axis defined by the unitary vector 

1 2 3 

a j l jSin, 

j 1,2,3 2 

consistent with (5), since: 

, 

 

a4 Cos , 

2 

(7) 

2 2 2 

l1l2l3 1 . (8) 

Then (6) simplifies to: 

2 

l1 Cos 

R l1l2l3Sin 

l1l3l2Sin l1l2 l3Sin 2 

l2 Cos l2l3 l1Sin l1l3 l2Sin 

 

l2l3 l1Sin , 

1 Cos 

, 

2 

l3 Cos 

 

 

(9) 

which is be studied in the present work. 

In next Section, the eigenvalue problem of (9) is used to write an R as a product of three 

matrices, which in turn explicitly shows the existence of a similarity transformation to 

diagonalize any rotation matrices. 

2. Eigenvalues and eigenvectors of R . 

The relation: 

R u u , 

(10) 

expresses the eigenvalue problem for the rotation matrix, such that is quite known [5] that: 

9 

(6)




l1 

1 

1 , u1 

l2 

, 

l 3 

e , e , 

i x i 

2 3 2 

however, it is very difficult to find explicitly 2 

following three vectors, proportional among them: 

u . In fact, the eigenvector 2 

2 2 

l2 l 3 l1l2 il3 l1l3 il2 

 

2 2 

l1l2il3 , 

 

l1 l3 

, 

 

l2l3 il1 

 

, 

2 2 

l1l3 il 

2 l2l3 il 

1 l1 l 

2 

10 

(11) 

u can be any of the 

and it is chosen as the most convenient according to the values of l1, l2 and l 3 , for instance, if 

l1 l3 0 , l2 

1, the second vector is the trivial solution u2 0 , and in that case it can be 

used any other vector of (12). Without losing generality, here it will be used: 

2 2 2 2 

l2 l 3 l2 l 

3 

 

u2 l1l2 il3 , u3 u2 l1l2 il3 

. 

l l il l l il 

1 3 2 1 3 2 

The problem of eigenvalues of a matrix gives algebraic information if it is also solved for the 

corresponding transpose matrix of (9) [6]: 

then getting the same eigenvalues (11), with: 

(12) 

(13) 

T 

R v v 

, 

(14)




 

1 1 

 

2 

 

2 

 

l1 

l1l2 il3 l1l2 il3 

v1 l2 , v 

2 , v 

 

2 2 3 

, 

2 2 

2l2l32l2l3 l 

3 

l1l3 il 2 l1l3 il 2 

 

2 2 2 2 

 

 

2l2l32l2l3 

which together with (8) and (13) imply: 

Then it is natural to construct the matrices: 

11 

(15) 

v u . 

(16) 

T 

j k jk 

2 2 2 2 

l1l2l3l2l 3 

 

U u1 u2 u3 l2l1l2il3l1l2il3 lllilllil 3 1 3 2 1 3 2 

100 Diag 

 

 

0 e 

0 

 

i 

e 

 

, 

i 

1 , 2, 3 

0 0 , 17 

1 1 

 

2 2 

 

l1 l1l2 il3 l1l2 il3 

V v1 , v2 , v3 l2 2 2 2 2 

2l2l32l2l3 l3 l1l3il2 l1l3 il 2 

 

2 2 

2 2 

2l2 

l3 

2l2 

l3 

 

 

and because the property (16) it results: 

V U 

T 1 

. 

(18) 

,




1 

U because when choosing 2 

Notice the existence of 

u given by (13) it has been supposed 

2 2 

that l2andl 3 do not simultaneously cancel, so that det U 2i l2 l3 

0 . 

General theorems of Linear Algebra [6] lead to the following factorization of the rotation 

matrix: 

(18) 

 

T 

 

1 

, 

(19) 

R U V U U 

which is easily seen through (8), (9) and (17). From (19) it is immediate that: 

12 

i i 

1 

U RU Diag e e 

1, , , (20) 

that is, the U matrix given explicitly by (17) generates a similarity transformation which 

diagonalizes R , q.e.d. 

References 

1. L. H. Ryder, Quantum field theory, Cambridge Univ. Press, Cambridge, England (1996) 

Chap. 2, ISBN: 0-521-47242-3 

2. E. Cartan, The theory of spinors, Dover, New York (1981) ISBN: 0-486-64070-1 

3. E. Piña G., A new parametrization of the rotation matrix, Am. J. Phys. 51 (1983) 375 

4. W. Rindler, What are spinors?, Am. J. Phys. 34, No.10 (1966) 937 

5. E. Piña G., Dinámica de rotaciones, Div. CBI, UAM-I, México DF (1996) Cap. 1, 

ISBN: 970-620-919-0 

6. C. Lanczos, Applied analysis, Dover, New York (1988) Chap. 2, ISBN: 0-486-65656-X

Una transformación de similaridad para la matriz de rotación - ijmes

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