On some trigonometric and hyperbolic functions evaluated on circulant matrices.

[section]1. Introduction

Given any sequence of numbers, [c.sub.0], [c.sub.1], ..., [c.sub.n - 1], we can form circulant matrices. From , circulant matrices have four types: the right circulant, the left circulant, the skew-right circulant and the skew-left circulant and take the following forms, respectively:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (1)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (2)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (3)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (4)

In each matrix, [??] = ([c.sub.0], [c.sub.1], ..., [c.sub.n - 1]) is called the circulant vector.

The said matrices have significant applications in different fields. These fields include physics, image processing, probability and statistics, number theory, geometry, numerical solutions of ordinary and partial differential equations , frequency analysis, signal processing, digital encoding, graph theory , and time-series analysis .

[section]2. Preliminaries

In this section, we shall use diag ([c.sub.0], [c.sub.1], ..., [c.sub.n - 1]) to denote the diagonal matrix

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

and adiag ([c.sub.0], [c.sub.1], ..., [c.sub.n - 1]) to denote the anti diagonal matrix

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Note that these matrices are not necessarily circulant.

Now, we define the Fourier matrix to establish the relationship of the circulant matrices.

Definition 2.1. The unitary matrix [F.sub.n] given by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (5)

where [omega] = [e.sup.2i[pi]/n] is called the Fourier matrix.

Here are the equations that show the relationship between the circulant matrices:

RCIR[C.sub.n]([??]) = [F.sub.n]D[F.sup.-1.sub.n], (6)

where [??] = ([c.sub.0], [c.sub.1], ..., [c.sub.n - 1]) and D = diag([d.sub.0], [d.sub.1], ..., [d.sub.n - 1]).

LCIR[C.sub.n]([??]) = [PI]RCIR[C.sub.n] ([??]), (7)

where [??] = ([c.sub.0], [c.sub.1], ..., [c.sub.n - 1]), D = diag([d.sub.0], [d.sub.1], ..., [d.sub.n - 1]), [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] matrix).

SRCIR[C.sub.n]([??]) = [DELTA][F.sub.n]D[F.sup.-1.sub.n]-1[[DELTA].sup.-1], (8)

where [??] = ([c.sub.0], [c.sub.1], ..., [c.sub.n - 1]), D = diag([d.sub.0], [d.sub.1], ..., [d.sub.n - 1]), [DELTA] = diag (1, [theta], ..., [[theta].sup.n - 1]), and [theta] = [e.sup.i[pi]/n].

SLCIR[C.sub.n]([??]) = [SIGMA]SRCIR[C.sub.n]([??]), (9)

where [??] = ([c.sub.0], [c.sub.1], ..., [c.sub.n - 1]), D = diag([d.sub.0], [d.sub.1], ..., [d.sub.n - 1]), [DELTA] = diag(1, [theta], ..., [[theta].sup.n - 1]), [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], [[??].sub.n - 1] = adiag (1, 1, ..., 1,1) (an (n - 1) * (n - 1) matrix), [O.sub.1] = (0,0, ..., 0,0).

The matrices [PI] and [SIGMA] are orthogonal meaning [PI] = [[PI].sup.T] = [[PI].sup.-1] and [SIGMA] = [[SIGMA].sup.T] = [[SIGMA].sup.-1] In case of right multiplication with these matrices, we have the following results from :

1. [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

2. [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

[section]2. Preliminary results

For the rest of the paper we shall use the following notations:

1. The set of complex right circulant matrices

RCIR[C.sub.n](C) = {RCIR[C.sub.n]([??])|[??] [member of] [C.sup.n]},

2. The set of complex left circulant matrices

LCIR[C.sub.n](C) = {LCIR[C.sub.n]([??])|[??] [member of] [C.sup.n]},

3. The set of complex s kew-right circulant matrices

SRCIR[C.sub.n](C) = {SRCIR[C.sub.n]([??])|[??] [member of] [C.sup.n]},

4. The set of complex s kew-left circulant matrices

SLCIR[C.sub.n](C) = {SLCIR[C.sub.n]([??])|[??] [member of] [C.sup.n]},

5. [e.sup.x] = E[x],

6. sin x = S[x],

7. cos x = C[x],

8. sinh x = Sh[x],

9. cosh x = Ch[x].

From , it has been shown that:

1. the sum of circulant matrices of the s ame type is a circulant matrix of the same type,

2. the product of right circulant matrices is a right circulant matrix,

3. the product of skew-right circulant matrices is a skew-right circulant matrix.

For the other products, we have the following lemmas which will be used to prove our results:

Lemma 2.1. The product of two left circulant matrices is a right circulant matrix. Proof.

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Lemma 2.2. The product of a left circulant and a right circulant is a left circulant matrix.

Proof. Case 1

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Case 2

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Lemma 2.3. The product of two skew-left circulant matrices is a skew-right circulant matrix.

Proof.

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Lemma 2.4. The product of a s kew-left circulant and a s kew-right circulant is a s kew-left circulant matrix.

Proof.

Case 1

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Case 2

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Lemma 2.5.

If k is odd, LCIR[C.sup.k.sub.n]([??]) [member of] LCIR[C.sup.k.sub.n](C); If k is even, LCIR[C.sup.k.sub.n]([??]) [member of] RCIR[C.sup.k.sub.n](C).

Lemma 2.6.

If k is odd, SLCIR[C.sup.k.sub.n] ([??]) [member of] SLCIR[C.sup.k.sub.n](C); If k is even, SLCIR[C.sup.k.sub.n]([??]) [member of] SRCIR[C.sup.k.sub.n](C).

[section]3. Main results

Theorem 3.1. E[RCIR[C.sub.n]([??])] is a right circulant matrix.

Proof.

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

It takes the form of Eq. 6, so it is a right circulant matrix.

Theorem 3.2. E[LCIR[C.sub.n]([??])] is a sum of right circulant matrix and a left circulant matrix.

Proof.

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

The first summand is a right circulant matrix because the powers of LCIR[C.sub.n]([??]) are even while the second summand is a left circulant matrix because the powers of LCIRC([??]) are odd.

Theorem 3.3. E[SRCIR[C.sub.n]([??])] is a skew-right circulant matrix.

Proof.

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

It takes the form of Eq. 8, so it is a skew-right circulant matrix.

Theorem 3.4. E[SLCIR[C.sub.n]([??])] is a sum of skew-right circulant matrix and a s kew-left circulant matrix.

Proof.

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

The first summand is a skew-right circulant matrix because the powers of SLCIR[C.sub.n]([??]) are even while the second summand is a skew-left circulant matrix because the powers of SLCIRC ([??]) are odd.

Theorem 3.5. S[RCIR[C.sub.n]([??])] is a right circulant matrix.

Proof.

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

It takes the form of Eq. 6, so it is a right circulant matrix.

Theorem 3.6. S[LCIR[C.sub.n]([??])] is a left circulant matrix.

Proof.

S[LCIR[C.sub.n]([??])] = [+[infinity].summation over k = 0] [(-1).sup.k] [LCIR[C.sup.2k + 1.sub.n]]/(2k + 1)!.

Since the powers of LCIR[C.sub.n] (vecc) are odd, it is a left circulant matrix.

Theorem 3.7. S[SRCIR[C.sub.n]([??])] is a skew-right circulant matrix.

Proof.

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

It takes the form of Eq. 8, so it is a skew-right circulant matrix.

Theorem 3.8. S[SLCIR[C.sub.n]([??])] is a skew-left circulant matrix.

Proof.

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Since the powers of SLCIR[C.sub.n] (vecc) are odd, it is a left circulant matrix.

Theorem 3.9. C[RCIR[C.sub.n]([??])] is a right circulant matrix.

Proof.

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

It takes the form Eq. 6, so it is a right circulant matrix.

Theorem 3.10. C[LCIR[C.sub.n]([??])] is a right circulant matrix.

Proof.

C[LCIR[C.sub.n]([??])] = [+[infinity].summation over k = 0] [(-1).sup.k] LCIR[C.sup.2k.sub.n]/(2k)!.

Since the powers of LCIR[C.sub.n]([??]) are even, it is a right circulant matrix.

Theorem 3.11. C[SRCIR[C.sub.n]([??])] is a skew-right circulant matrix.

Proof.

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

It takes the form Eq. 8, so it is a right circulant matrix.

Theorem 3.12. C[SLCIR[C.sub.n]([??])] is a skew-right circulant matrix.

Proof.

C[LCIR[C.sub.n]([??])] = [+[infinity].summation over k = 0] [(-1).sup.k] SLCIR[C.sup.2k.sub.n]/(2k)!.

Since the powers of SLCIR[C.sub.n]([??]) are even, it is a skew-right circulant matrix.

Theorem 3.13. Sh[RCIR[C.sub.n]([??])] is a right circulant matrix.

Proof.

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

It takes the form of Eq. 6, so it is a right circulant matrix.

Theorem 3.14. Sh[LCIR[C.sub.n]([??])] is a left circulant matrix.

Proof.

Sh[LCIR[C.sub.n]([??])] = [+[infinity].summation over k = 0] [LCIR[C.sup.2k + 1.sub.n]([??])]/(2k + 1)!.

Because the powers of LCIR[C.sub.n]([??]) are odd, it is a left circulant matrix.

Theorem 3.15. Sh[SRCIR[C.sub.n]([??])] is a skew-right circulant matrix.

Proof.

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

It takes the form of Eq. 8, so it is a skew-right circulant matrix.

Theorem 3.16. Sh[SLCIR[C.sub.n]([??])] is a skew-left circulant matrix.

Proof.

Sh[SLCIR[C.sub.n]([??])] = [+[infinity].summation over k = 0] [SLCIR[C.sup.2k + 1.sub.n] ([??])]/(2k + 1)!.

Because the powers of SLCIR[C.sub.n]([??]) are odd, it is a skew-left circulant matrix.

Theorem 3.17. Ch[RCIR[C.sub.n]([??])] is a right circulant matrix.

Proof.

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

It takes the form of Eq. 6, so it is a right circulant matrix.

Theorem 3.18. Ch[LCIR[C.sub.n]([??])] is a right circulant matrix.

Proof.

Ch[LCIR[C.sub.n]([??])] = [+[infinity].summation over k = 0] [LCIR[C.sup.2k.sub.n]]/(2k)!.

Because the powers of LCIR[C.sub.n]([??]) are even, it is a right circulant matrix.

Theorem 3.19. Ch[SRCIR[C.sub.n]([??])] is a skew-right circulant matrix.

Proof.

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

It takes the form of Eq. 8, so it is a skew-right circulant matrix.

Theorem 3.20. Ch[SLCIR[C.sub.n]([??])] is a skew-right circulant matrix.

Proof.

Ch[SLCIR[C.sub.n]([??])] = [+[infinity].summation over k = 0][SLCIR[C.sup.2k.sub.n]]/(2k)!.

Because the powers of SLCIR[C.sub.n]([??]) are even, it is a skew-right circulant matrix.

[section]4. Conclusion

Right and skew-right circulant matrices remains invariant on their type when evaluated on trigonometric and hyperbolic functions that are used in this paper. On the other hand, the left and skew-left circulant matrices change their type depending on the parity of the trigonometric and hyperbolic function.

References

 A. Bueno, Operations and inversions on circulant matrices, masters thesis, University of the Philippines Baguio.

 H. Karner, J. Schneid and C. Ueberhuber, Spectral decomposition of real circulant matrices, Linear Algebra and Its Applications, 367(2003), 301-311.

 D. Pollock, circulant matrices and time-Series analysis, Queen Mary, University of London, August 2001.

 A. Wyn-Jones, Circulant s, Carlisle, Pennsylvania.

Aldous Cesar F. Bueno

Department of Mathematics and Physics, Central Luzon State University Science City of Munoz 3120, Nueva Ecija, Philippines