Re: The cosine of a matrix

On Wed, 25 Oct 2006 00:50:37 GMT, "*** T. Winter" <***.Winter@xxxxxx>
wrote:

In article <ehlbla$4a9$00$1@xxxxxxxxxxxxxxxxx> Gottfried Helms <helms@xxxxxxxxxxxxx> writes:
Am 24.10.2006 17:20 schrieb C6L1V@xxxxxxx:
schoenfeld.one@xxxxxxxxx wrote:
In this article we shall derive an explicit formula for the cosine of a
matrix.

All if this is old hat and well-known; see, eg., Gantmacher, or
Lancaster.

I have a related question, though: is it true that for a matrix A we
have
(sin(A))^2 + (cos(A))^2 = I? (I = identity matrix)

I think, this is simple:
it sums the (diagonal matrix of) eigenvalues to 1 and leads
by
E *cos(D)²*E^-1 + E *sin(D)²*E^-1

If the matrix is not defective (i.e. is diagonisable). Off-hand I
do not know whether it is also valid for defective matrices (I
suspect not).

Surely it's ok for any square matrix. For example, the diagonalizable
matrices are dense.

(Or if you like you can give harder proofs. For example, the fact
that the identity holds for complex numbers implies that the
coefficients in the power series for sin and cos satisfy
certain equations, and those equations imply the identity for
any matrix, or for that matter any element of a Banach algebra.)

The original poster ignores the possibility where the two eigenvalues are
equal. His formulas do not even work for the identity matrix.

On the other hand, if a matrix is diagonisable, it can be written as
M = U.D.V
where D is a diagonal matrix and where V.U = I. It is easy to show that
cos(M) = U.cos(D).V
and from that that cos(M)^2 + sin(M)^2 = I.

Somebody else questioned the usability. It has its uses in sets of
differential equations.


************************

David C. Ullrich
.

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