The classical(geometrical) definition of is as follows:
For an angle (that is for a real number that lies between 0 and ) is the ratio
where is the opposite side and is the hypotenuse of the right triangle with the angle . \hr
Clearly, this definition cannot be expanded to incorporate matrices or complex numbers instead of angles.
Also, the function can be defined by the Taylor series:
sin of a complex argument
Now one can compute of any real or complex number.
can easily prove the selebrated Euler's formula
Notice that does not need to be smaller than
one any more; on the contrary, the equation
sin of matrices
Recall, that the Taylor series uses only a sequence of additions and multiplications
to define a function. These operations are well defined for any square
matrix , where is an arbitrary integer.
Therefore, we have
Similarly, we can define any other analytic function of a square matrix :