Circulant Matrices. Ashley Lorenz

Transcription

1 irculant Matrices Ashley Lorenz Abstract irculant matrices are a special type of Toeplitz matrix and have unique properties. irculant matrices are applicable to many areas of math and science, such as physics and di erential equations. They are also useful in digital image processing. They appear in Sudoku puzzles and other types of Latin squares. A n x n circulant matrix is formed by cycling its entries until (n-) new rows are formed. irculant matrices share a relationship with a special permutation matrix,, and can be expressed as a linear combination using powers of. The roots of unity play an important role in the calculation of the eigenvalues and can be used to compute eigenvalues of any circulant matrix. irculant matrices also have other special properties such as their ability to commute with one another. This article describes these properties in detail. The rst section describes the general circulant matrix, A; and its relationship with a special permutation matrix,. Knowing the relationship that these matrices hold is enough to write A as a linear combination of the powers of. Through examples, we obtain a formula for the general circulant matrix which is useful for the calculation of the eigenvalues as described later. De nition irculant Matrix A matrix A M n (R) is called a circulant matrix if it has the form below:

2 A = a a a 3 ::: ::: a n a n a n a a ::: ::: a n a n a n a n a ::: ::: a n 3 a n... ::: :::.. B... ::: a 3 a 4 a 5 ::: ::: a a A a a 3 a 4 ::: ::: a n a Note: Notice how the entries of a circulant matrix are generated. The entries in the rst row consist of a through a n : In the second row, a n takes the place of a and shifts the remaining entries to the right one position. This process is continued for the entire matrix, in a cycling pattern. The diagonals have constant entries, making it a type of Toeplitz matrix. De nition Permutation Matrix A permutation matrix can be de ned as any matrix with a single one in each row and column with the remaining entries zeros. The matrix M n (R) given below is an example of a permutation matrix and shares a special relationship with a circulant matrix, A. :::. :::. = ::: ::: ::: A ::: ::: Ones appear on the upper diagonal and in the lower left hand corner of this matrix. The matrix itself is also a circulant matrix, shifting the ones to the right one position for the remaining (n-l) rows. Exercise 3 Let A be a circulant matrix. Let be a permutation matrix. Find a relationship between A and the powers of.

3 a a A = a a = Square the matrix : Squaring yields the identity matrix. = = = I Multiplying a by I and adding it together with the product of a and forms A in its entirety. a a A = a + a = + = a a a a a a A = a I + a A can be written as a linear combination using the powers of. Exercise 4 Let A be a 3 3 circulant matrix. Let be a 3 3 permutation matrix. Find a relationship between A and the powers of. a a a 3 A a 3 a a A A a a 3 a Square the matrix : Squaring yields a new permutation matrix with the ones shifted to the right one position. A A Raise to the third power. Raising to the third power yields the identity matrix, I: 3 A A = I Multiplying a by I and adding it together with the product of a and and a 3 and forms A in its entirety. 3

4 A = A + A + a A a a a 3 a A a A a 3 A a a a 3 a a a 3 a 3 a a A a a 3 a A = a I + a + a 3 A can be written as a linear combination using the powers of. Exercise 5 Let A be a 4 4 circulant matrix. Let be a 4 4 permutation matrix. Find a relationship between A and the powers of. a a a 3 a 4 A = Ba 4 a a a a 3 a 4 a a A = A a a 3 a 4 a The same process is completed for the 4 4 case of the matrix. = B A = A 3 = B A = A 4 yields the identity matrix. 4 = B A = A = I 4

5 A = a A + a A + a B A + a B A a a a 3 a 4 = B a A + B a A + B a a 3 A + Ba a 4 A a a a 3 a 4 a a a 3 a 4 = Ba 4 a a a a 3 a 4 a a A a a 3 a 4 a A = a I + a + a 3 + a 4 3 Raising to the,, and 3 respectively, provides the matrices necessary to write A as a linear combination of the powers of : Notice how in the previous three examples, raising to the n yielded the identity matrix. Each a j component can be multiplied by its corresponding matrix, always one less degree than its subscript. From this idea, the following proposition can be formulated. Proposition 6 Let A M(R). Then A is circulant i A = P n k= a k+ k ; where is the permutation matrix. Proof It is easy to observe that raising to the f; ; ; 3:::ng provides the matrices necessary to produce A as a linear combination of the powers of : ::: ::: ::: ::: ::: ::: ::: ::: = ; = ; A ::: ::: ::: ::: A ::: ::: ::: ::: ::: ::: ::: 5

6 a : ::: ::: = ; ::: ; n = I A ::: ::: ::: The a terms are generated by multiplying the identity matrix by In a circulant matrix, the a s always fall on the diagonal. ::: ::: ::: a ::: ::: :::.... ::: ::: ::: a.... ::: ::: :::.. ::: :::. a... ::: ::: a = B. A a The a terms are generated by multiplying by a : ::: ::: ::: a ::: ::: ::: ::: ::: a ::: ::: ::: ::: a.... a ::: ::: = B a A ::: ::: ::: ::: a ::: ::: ::: ::: Notice how the s in correspond to the a positions in A: The a 3 terms are generated by squaring and multiplying it by a 3 : 6

7 a 3 ::: a 3 ::: a = B.... B.... a ::: ::: ::: ::: a 3 ::: ::: ::: ::: A ::: ::: ::: a 3 ::: ::: ::: The s in the matrix correspond to the a 3 positions in A: Repeating this process for a 4 through a n all of the a j terms can be generated by mutlipying by powers of : For instance, a 4 3 generates the a 4 positions, a 5 4 generates the a 5 positions, etc. Once all of these matrices are created, they can be added together to form A in its entirety. A = a a a 3 ::: ::: a n a n a n a a ::: ::: a n a n a n a n a ::: ::: a n 3 a n... ::: :::.. = a I + a + a 3 + a ::: + a n n B... ::: a 3 a 4 a 5 ::: ::: a a A a a 3 a 4 ::: ::: a n a Xn A = a k+ k I Equation k= ROOTS OF UNITY A brief description of the roots of unity is necessary for the computation of the eigenvalues. Roots of unity consist of all the z values, both real and complex, that satisfy the following equation: z n = The value of n will correspond with the number of roots of unity. For instance, when n = 3 there will be 3 roots of unity. The roots of unity are useful for the calculation of eigenvalues of any 7

8 permutation matrix. The solutions to the equation above can be found by using the following equation: w = cos + i sin These numbers, w, lie on the unit circle and can be generated by substituting = k; where k is an integer f; ; 3:::ng. n = is called the "primitive root of unity" because it is the angle n that lies between each root of unity and brings the roots back to the start of the unit circle. When added, the roots of unity (when n >), equal zero. The complex plane consists of both real and imaginary numbers. In general, z = x + iy: Using polar coordinates, we obtain w. r = p x + y = jzj z = (x; y) = (r cos ; r sin ) So, z = r cos + ir sin : Euler s formula states that cos + i sin = e i z = r(e i ) z = r(cos + i sin ) Using w = cos + i sin ; we can nd the roots of unity for any n-value. Example 7 Find the roots of unity for z = Although the solution is obviously, the formula illustrates it well. Since n =, = w = cos + i sin = Example 8 Find the roots of unity for z = Since n =, = and The angle between the two roots of unity is ; the primitive root of unity. w = cos + i sin = w = cos + i sin = 8

9 Example 9 Find the roots of unity for z 3 = Since n = 3, = 3 ; 4 3 and w = cos 3 + i sin 3 = +p 3i w = cos i sin 4 3 = p 3i w = cos + i sin = The angle between the roots of unity is 3 : Example Find the roots of unity for z 4 = Since n = 4, = ; ; 3 ; and w = cos + i sin = i w = cos + i sin = w = cos 3 + i sin 3 = i w = cos + i sin = The angle between the roots of unity is : Example Find the roots of unity for z 6 = Since n = 6, = ; ; ; 4; 5 ;and w = cos 3 + i sin 3 = +p 3i w = cos 3 + i sin 3 = +p 3i w = cos + i sin = w = cos i sin 4 3 = p 3i w = cos i sin 5 3 = p 3i w = cos + i sin = The angle between the roots of unity is 3 : Example Find the roots of unity for z 8 = Since n = 8, = ; ; 3; ; 5; 3; 7 ; and w = cos 4 + i sin 4 = p + p i w = cos + i sin = i 9

10 w = cos i sin 3 4 = p + p i w = cos + i sin = w = cos i sin 5 4 = p p i w = cos 3 + i sin 3 = i w = cos i sin 7 4 = p p i w = cos + i sin = The angle between the roots of unity is 4 : Summary 3 z n = would have n roots of unity that are evenly spaced by an angle of around the unit circle. Also, the sum on n the roots of unity (excluding when z = ) always equals zero. EIGENVALUES OF A IRULANT MATRIX The roots of unity play an important role in determining the eigenvalues of a circulant matrix. Finding the eigenvalues of a permutation matrix, and using a polynomial corresponding to Equation can compute the eigenvalues of any circulant matrix, A. A de nition of an eigenvalue is needed to understand the procedure. Also, de nitions of the characteristic polynomial and trace are provided to see the relationship the roots of unity play with these ideas. De nition 4 For B M n there exists a non-zero vector x n such that Bx = x; for some scalar : Here is the eigenvalue of B and x 6= is the eigenvector that is associated with the particular : We can consider as an eigenvalue of B if and only if the det(i B) = : The set of all eigenvalues of B is called the spectrum, written as (A) = f j is an eigenvalue of Bg : De nition 5 The characteristic polynomial of a matrix B M n is described as p B (t) = det(ti B): De nition 6 The trace of B is the sum of the eigenvalues and is denoted as tr(b). Example 7 Find the eigenvalues of = :

11 Set the det(i ) = = det = = = = () = f; g Notice that the eigenvalues of are the roots of unity for z = : p (t) = det(ti ) = t = (t )(t + ) tr() = tr() = is consistent with the idea that the sum of the roots of unity equals zero, since the eigenvalues of are the roots of unity for z = : Example 8 Find the eigenvalues of A : A =

12 ( ) = 3 = 3 = = ; ( () = ; p 3i + p 3i ; p ) 3i Notice that the eigenvalues of are the roots of unity for z 3 = : p (t) = t 3 " = (t ) t + p!# " 3i t p!# 3i tr() = Example 9 Find the eigenvalues of = A : det A = ( 3 ) = 4 = = ; i () = f; ; i; ig p (t) = t 4 = (t )(t + )(t i)(t + i)

13 onclusion A permutation matrix will have n distinct eigenvalues that satisfy z n =, the roots of unity. These eigenvalues are the zeros of the characteristic polynomial. By de nition, the eigenvalue equation is x = x for some x 6= : (Note: This is true for any matrix B M n but here we are interested in the permutation matrix and its relationship to a circulant matrix, A.) Using a (n-) th polynomial, p(t), we can substitute the permutation matrix,, into the polynomial. p(t) = a + a t + a 3 t + ::: + a n t n is a n-l th degree polynomial. Likewise, p() = a I + a + a 3 + ::: + a n n by Equation : The introduction of a proposition with proof is needed. Proposition If is an eigenvalue of a matrix then n is an eigenvalue of n with the same corresponding eigenvector x 6= :(Note: This is true for any matrix B M n, but here we are only interested in the matrices and A.) Proof. x = x ) x =(x) ) x =(x) ) x =(x) ) x = x ( n )x = n x ) is an eigenvalue of with eigenvector x 6= : Similarly, this proof can be extended to the n case. ) n x =( n x) ) n x =( n x) ) n x = n x ) n has an eigenvalue of n with eigenvector x 6= : 3

14 Proposition Let p(t) = a + a t + a 3 t + :::+ a n t n be an (n-) th degree polynomial. Suppose is an eigenvalue of with corresponding eigenvector x. Then p() is an eigenvalue of p() with the same eigenvector x. Proof. p() = A = a I + a + a 3 + ::: + a n n p()x = (a I + a + a 3 + ::: + a n n )x = a x + a x+ a 3 x+::: + a n n x = (a + a + a 3 +::: + a n n )x ) P ()x = P ()x ) P () is an eigenvalue of the matrix polynomial P () with eigenvector x 6= : Using this proof for the previous examples with circulant matrices A ; A 33 ; A 44; we can easily obtain the eigenvalues of these matrices. a a Example 3 Find the eigenvalues of A = : a a A = a I + a p(t) = a + a t Evaluate p(t) at the eigenvalues of the corresponding matrix. = ) p() = a + a = ) p( ) = a a Using real numbers, we can extend this idea to a similar matrix; D = 3 4 : 4 3 p(t) = 3 + 4t = ) p() = 7 = ) p( ) = The eigenvalues of D are = 7 and = 4

15 a a a 3 Example 4 Find the eigenvalues of A a 3 a a A : a a 3 a Using the idea from above, we can evalute p(t) using the roots of unity for z 3 = for the 3 3 case. A = a I + a + a 3 p(t) = a + a t + a 3 t = ) p() = a + a + a 3 = + p 3i + p! 3i ) p + p! 3i + p 3i = a + a + a 3 3 = p p! 3i 3i ) p p! p 3i 3i = a + a + a 3 The eigenvalues of A are!! = a + a + a 3 ; + p! p! 3i 3i = a + a + a 3 ; p! 3i + p! 3i and 3 = a + a + a 3 : 3 For example, we can extend this idea to a similar matrix; E 3 A 3 5

16 p(t) = + t + 3t = ) p() = 6 = + p 3i + p! 3i ) p = + p! 3i + p 3i = = p p! 3i 3i ) p p! p 3i 3i = The eigenvalues of E are = 6; = 3 3i and 3 = p 3 5p 3i a a a 3 a 4 Example 5 Find the eigenvalues of A = Ba 4 a a a a 3 a 4 a a A : a a 3 a 4 a!! Using the same principle, the idea can be stemmed to the 4 4 case. A = a I + a + a 3 + a 4 3 p(t) = a + a t + a 3 t + a 4 t 3 = ) p() = a + a + a 3 + a 4 = ) p( ) = a a + a 3 a 4 3 = i ) p(i) = a + a i a 3 a 4 i 4 = i ) p( i) = a a i + a 3 + a 4 i The eigenvalues of A are = a + a + a 3 + a 4 ; = a a + a 3 a 4 ; 3 = a + a i a 3 a 4 i; and 4 = a a i + a 3 + a 4 i 4 3 For example, we can extend this idea to a similar matrix; F = B 4 4 3A 3 4 6

17 p(t) = 4 + 3t + t + t 3 = ) p() = = ) p( ) = 3 = i ) p(i) = + i 4 = i ) p( i) = 6 i onclusion 6 A circulant matrix A M n has a representative (n-) th degree polynomial, p(t) = a +a t+a 3 t +:::+ a n t n. Using the roots of unity satisfying the equation z n = ; this polynomial can be used to nd the corresponding eigenvalues of A with eigenvector x 6= : irculant matrices have special properties that tie to other areas of mathematics, making it an interesting topic to study. Its relationship with a permutation matrix,, allow it to be written as a linear combination using the powers of : Understanding the roots if unity makes it easy to compute the eigenvalues of. Using those roots, it is easy to calculate the eigenvalues of any circulant matrix with the help of a polynomial corresponding to Equation. AKNOWLEDGEMENT I would like to thank Dr. Bathi Kasturiarachi for his guidance through the writing of this article, for he has taught me all that I know about matrix theory and its applications. REFERENES [] HORN, R. A. and JOHNSON,. R., 985, Matrix Analysis (ambridge University Press). [] LIPSHUTZ, S. and LIPSON, M.,, Linear Algebra, 3rd edn (McGraw Hill). [3] STRANG, G., 6, Linear Algebra and Its Applications, 4th edn (Thomson/Brooks/ole). 7