Some explicit formulas for the matrix exponential

Author

Bernstein, Dennis S. ; So, Wasin

Author_Institution

Dept. of Aerosp. Eng., Michigan Univ., Ann Arbor, MI, USA

Volume

38

Issue

8

fYear

1993

fDate

8/1/1993 12:00:00 AM

Firstpage

1228

Lastpage

1232

Abstract

Formulas are derived for the exponential of an arbitrary 2×2 matrix in terms of either its eigenvalues or entries. These results are then applied to the second-order mechanical vibration equation with weak or strong damping. Some formulas for the exponential of n×n matrices are given for matrices that satisfy an arbitrary quadratic polynomial. Besides the above-mentioned 2×2 matrices, these results encompass involutory, rank 1, and idempotent matrices. Consideration is then given to n×n matrices that satisfy a special cubic polynomial. These results are applied to the case of a 3×3 skew symmetric matrix whose exponential represents the constant rotation of a rigid body about a fixed axis

Keywords

eigenvalues and eigenfunctions; matrix algebra; polynomials; 2×2 matrices; 3×3 skew symmetric matrix; constant rotation; cubic polynomial; eigenvalues; explicit formulas; idempotent matrices; involutory matrices; matrix exponential; quadratic polynomial; rank 1 matrices; rigid body; second-order mechanical vibration equation; strong damping; weak damping; Control systems; Control theory; Differential equations; Eigenvalues and eigenfunctions; Kinematics; Linear systems; Pervasive computing; Quaternions; Symmetric matrices; Vectors;

fLanguage

English

Journal_Title

Automatic Control, IEEE Transactions on

Publisher

ieee

ISSN

0018-9286

Type

jour

DOI

10.1109/9.233156

Filename

233156