San José State University

applet-magic.com
Thayer Watkins
Silicon Valley
USA

The Exponential Matrix
Function of a Purely
Imaginary Matrix and the
Trigonometric Identities

The matrix exponential function of an m×m matrix M is defined as

#### Exp(M) = I/0! + M/1! + M²/2! + … + Mn/n! + …

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

#### Exp(iQ) = Cos(Q) + i·Sin(Q)

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

#### Lemma 0: 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

#### Lemmas 2 and 3: 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

#### exp(A)·exp(B) = exp(A+B) only if AB = BA

In the above this condition is trivially satisfied.

Now consider [exp(A)]n for n being a nonnegative integer. Then

Then

#### Lemma 6: Cos²(Q) = [Cos(2Q) +I]/2

Since Cos(Q)=[Exp(iQ) + Exp(−iQ)]/2

Likewise

#### Lemma 7: Sin²(Q) = [I − Cos(2Q)]/2

Since Sin(Q)=[Exp(iQ) − Exp(−iQ)]/2i

#### Sin²(Q) = {[Exp(iQ)]² − 2Exp(iQ)·Exp(−iQ) + [Exp(−iQ)]²}/(−4) which reduces to Sin²(Q) = [Exp(2iQ) + Exp(−2iQ) − 2I]/(−4) and further to Sin²(Q) = [I −Cos(2Q)]/2

Consider Sin(Q)Cos(Q). This is

Thus

And likewise

And furthermore

Now define

#### Tan(Q) = [Cos(Q)]−1Sin(Q)

and then consider

#### Cos²(Q)[I + Tan²(Q)]

This reduces successively to

Thus

#### Lemma 10: 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)

#### Lemma 11: 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

#### Lemma 12: Exp(iπI) = −I

(To be continued.)