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Function of a Purely Imaginary Matrix and the Trigonometric Identities |
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The matrix exponential function of an m×m matrix M is defined as
where I is the m×m identity matrix. The series converges for all M.
If M is purely imaginary, such as iQ where the elements of Q are all real, then
The topic pursued here is the proof that the matrix cosine and sine functions satisfy the common trigonometric identities, such as cos²(φ)+sin²(φ)=1.
First note
If O is any square matrix of all zeroes then
Exp(O) = I
Putting O into the defining series results in all terms except the first being equal to O.
Then note that the complex conjugate of Exp(iQ) is Exp(−iQ). Thus
Thus
Cos²(Q) + Sin²(Q) = I
Cos(Q) = [Exp(iQ) + Exp(−iQ)]/2
Sin(Q) = [Exp(iQ) − Exp(−iQ)]/(2i)
In dealing with the product of exponential care must be taken because
In the above this condition is trivially satisfied.
Now consider [exp(A)]^{n} for n being a nonnegative integer. Then
[exp(A)]^{n} = exp(nA)
Then
Cos²(Q) = [Cos(2Q) +I]/2
Since Cos(Q)=[Exp(iQ) + Exp(−iQ)]/2
Likewise
Sin²(Q) = [I − Cos(2Q)]/2
Since Sin(Q)=[Exp(iQ) − Exp(−iQ)]/2i
Consider Sin(Q)Cos(Q). This is
Thus
Sin(Q)Cos(Q) = Sin(2Q)/2 = [I − Cos(2Q)]/2
And likewise
Cos(Q)Sin(Q) = Sin(2Q)/2 = [I − Cos(2Q)]/2
And furthermore
Sin(Q)Cos(Q) = Cos(Q)Sin(Q)
Now define
and then consider
This reduces successively to
Thus
I + Tan²(Q) = [Cos²(Q)]^{−1} = [Cos(Q)^{−1}]²
where Cos(Q)^{−1} could be denoted as Sec(Q).
Similarly if Cot(Q)=Sin^{−1}(Q)Cos(Q)
I + Cot²(Q) = [Sin²(Q)]^{−1} = [Sin(Q)^{−1}]²
where Sin(Q)^{−1} could be denoted as Csc(Q).
Finally to make things complete
Exp(iπI) = −I
(To be continued.)
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