matrix exponential
The exponential of a real valued square matrix , denoted by , is defined as
Let us check that is a real valued square matrix. Suppose is a real number such for all entries of . Then for all entries in , where is the order of . (Alternatively, one could argue using matrix norms: We have for the 2-norm, and hence the entries of are bounded by .) Thus, in general, we have . Since converges, we see that converges to real valued matrix.
Example 1. Suppose is nilpotent, i.e., for some natural number . Then
Example 2. If is diagonalizable, i.e., of the form , where is a diagonal matrix, then
Further, if , then whence
For diagonalizable matrix , it follows that . However, this formula is, in fact, valid for all .
Let be a square real valued matrix. Then the matrix exponential satisfies the following properties
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1.
For the zero matrix , , where is the identity matrix.
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2.
If for an invertible matrix , then
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3.
If and commute, then .
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4.
The trace of and the determinant of are related by the formula
In effect, is always invertible. The inverse is given by
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5.
If is a rotational matrix, then .
A relevant example on property 3.
We report an interesting example where the cited property is valid. In the field of complex numbers consider the complex matrix
(1) |
being hermitian, i.e. (here ”” and overline ”” stand for tranposition and conjugation, respectively) and orthogonal, i.e . From (1),
Since is orthogonal, from the complex equation ( is the identity matrix), we have
whence the imaginary part leads to the equation
(2) |
But is also hermitian, so that
therefore is symmetric, and is skew-symmetric. From these and (2), , and this implies that . So that, the real and imaginary parts of an orthogonal and hermitian matrix verifies the property. Likewise, it is easy to show that if the complex matrix is symmetric and unitary, its real an imaginary components also verify this property.
Title | matrix exponential |
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Canonical name | MatrixExponential |
Date of creation | 2013-03-22 13:33:27 |
Last modified on | 2013-03-22 13:33:27 |
Owner | mathcam (2727) |
Last modified by | mathcam (2727) |
Numerical id | 13 |
Author | mathcam (2727) |
Entry type | Definition |
Classification | msc 15A15 |
Classification | msc 15-00 |
Related topic | ProofOfEquivalenceOfFormulasForExp |