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.
Further, if , then whence
If for an invertible matrix , then
If and commute, then .
If is a rotational matrix, then .
Since is orthogonal, from the complex equation ( is the identity matrix), we have
whence the imaginary part leads to the equation
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.
|Date of creation||2013-03-22 13:33:27|
|Last modified on||2013-03-22 13:33:27|
|Last modified by||mathcam (2727)|