1. Peter Loly with thanks to George Styan, Ian Cameron
Two* small theorems for square matrices rotated a quarter turn. (*a third added just before the meeting ) WCLAM2008 May 30-31, 2008 (rev. 2,3 June, 2008)
Peter Loly
with thanks to
George Styan, Ian Cameron
I am glad to have been invited by George Styan to talk to this group, and lead off a session on magic squares. This may be a pivotal event in what I like to call “The Science of Magical Squares”. Ian Cameron is my co-author for this talk and we both hope to begin fruitful discussions with many of you during the workshop. We are also to learn of your interests.

2. Abstract
The eight "phases" of a general square matrix resulting from rotations and reflections may be separated into two sets which interlace each other (alpha, beta).
The effects of rotation (e.g. a quarter turn) have the effect of moving between the two sets.
We have found two theorems which describe:
1) sign changes of the determinant (if non-singular), and
2) a pair of characteristic equations (and thus eigenvalues).

3. Second order example 1 2 3 4 eigenvalues: (5/2)±(1/2)√(33)
characteristic polynomial: X²-5X-2
eigenvalues:
(5/2) ±(1/2)√(17)
characteristic polynomial: X²-5X+2 3 1 4 2

4. J
The reversal operator of all ones along the dexter diagonal is used in the proofs of these theorems. Simple second order examples suffice to illustrate the theorems, which emerged from a study of higher order matrices.

5. Reversal matrix, Jn (e.g. n=6)1

6. dihedral group of order 8, D4 (R4 =I ,T2 =I, T=J, and prime ‘ for transpose)
I (A) α
R (A) β
R2 (A) α
R3 (A) β
R T (A) α
R2 T (A) β
R3 T (A) α
T (A) β A
(JA)’ =A’J
JAJ
(AJ)’ =JA’ A’ AJ
JA’J JA

7. Rotate CW A B C D C A D B D C B A B D A C

8. After top-bottom flip over, rotate CWD B A D C D B C A C D A B

9. 8 phases shown on a cube
They may be indicated on the front plane and the back plane of a 2-by-2-by-2 cube:
Front: A, A′J, JAJ, JA′
Back: A′, JA, JA′J, AJ
Note that the clockwise sequence on the front plane A,A′J,JAJ,JA′ continues clockwise on the back plane A′,JA,JA′J,AJ before returning to the front face. Also a transpose about the second diagonal is given by JA′J.
Alpha set: A, JAJ, A’, JA’J; Beta: A’J,JA’,JA,AJ.
N.B. Styan et al use “sweet” and “sour”.

10. Clockwise lower left on Front face, then upper right on Back face.α AJ β JA
JA’J A α
A’J β
JA’
JAJ

11. Theorem 1: Effect of rotation on the determinant
The determinant of an nth order nonsingular square matrix rotated by a quarter turn remains of constant magnitude with its sign varying between successive phases as the determinant of the corresponding nth order reversal matrix:
det(Jn)=(-1)int(n/2) , n≥2.

12. Proof of Theorem 1
Since the determinant is unchanged under transpose, and the determinant of the matrix products is the product of their determinants, the determinant of a matrix phase involves either:
a factor of det(Jn), and thus alternating sign changes,
or its square, and thus no sign changes for the latter set.

13. Proof of Theorem 1 - continued
Let n be the order of the square matrix, then det(Jn) is bialternant(?):
det(J2)=(-)1, det(J3)=(-)1, det(J4)=(+)1, det(J5)=(+)1, det(J6)=(-)1, etc. This is easily seen by multiplying out the determinant and observing that the number of multiplied negative signs is the integer part of (n/2).
So the sequence of signs runs in -- and ++ pairs as n increases.

14. Reversal matrix, Jn 1 1 1 1

15. Rotation and Determinant
Sign change of determinant for
n mod 4 > 1 (or for n = 2 + 4r and n = 3 + 4r, r = 0,1, 2..), i.e., for n = 2, 3; 6, 7; 10,11; ..
No sign change for n = 4,5; 8,9; 12,13, ..

16. Theorem 2: Two characteristic polynomials and eigenvalue sets
Under D4 there are two alternating characteristic polynomials,
CPα: det(A-xI)=0, the “usual” case,
the other, CPβ: det(A-yJ)=0,
Both can be obtained from the more general: det(A-xI-yJ).
The difference is simply one diagonal or the other!

17. Proof of Theorem 2 Beta: A′J, JA′, JA, AJ
have the diagonals swapped compared to alpha: A, JAJ, A′, JA′J.
From a CPβ: det(JA-xJ)=det(J)det(A-xI)
=± det(A-xI); But sign of det(J) is irrelevant: CPα.
and similarly from a CP α to a CPβ:
det(J)det(A-xI)/det(J)=det(JA-xJ)/ det(J)
=± det(JA-xJ), …

18. Bonus Theorem 3: Rotation and SVD
Use the symmetric A’A (or AA’):
cpSVDα=det(A’A-xI) (α)
Pre- and post-multiply (α) by det(J): det(J)det(A’A-xI)=det[J(A’A)-xJ],
then det[J(A’A)-xJ]det(J)=det[(JA’)(AJ)-xI],
and finally =det[(AJ)’(AJ)-xI], since (AJ)’=JA’,
In which AJ is from the β set, so there is no change of SVD values on rotation.

19. References
Hruska, F., Magic Squares, Matrices, Planes and Angles, Journal of Recreational Mathematics 23 (1991)
van den Essen, A., Magic squares and linear algebra, American Mathematical Monthly 97 (1990)
Loly, Cameron, Trump and Schindel, Magic Square Spectra, IWMS16, 2007 Windsor, submitted for publication.
K. L. Chu, G. P. H. Styan, G. Trenkler & K. Vehkalahti, Some comments on magic squares, with special emphasis on magic matrices having 3 nonzero eigenvalues, In preparation, August 2007.
A.C. Cayley, Note on Magic Squares, The Messenger of Mathematics, VI (1877) 168.

20. finis
Thank You

21. 35 48 3 1 6 40 42 19 34 28 21 20 46 7 11 26 38 13 45 33 9 18 36 27 25 23 14 32 41 17 5 37 12 24 39 43 4 30 29 22 16 31 8 10 44 49 47 2 15 15 12 21 10 7 2 6 17 18 22 25 23 13 3 1 4 8 9 20 24 19 16 5 14 11 8 1 6 3 5 7 4 9 2 64 9 17 40 32 41 49 8 2 55 47 26 34 23 15 58 3 54 46 27 35 22 14 59 61 12 20 37 29 44 52 5 60 13 21 36 28 45 53 4 6 51 43 30 38 19 11 62 7 50 42 31 39 18 10 63 57 16 24 33 25 48 56 1 2 9 4 29 36 31 34 32 30 7 5 3 6 1 8 33 28 35 20 27 22 11 18 13 25 23 21 16 14 12 24 19 26 15 10 17 16 3 2 13 5 10 11 8 9 6 7 12 4 15 14 1
1693 Frenicle counted 880 in order 4; 1972 Richard Schroeppel used a computer to obtain some 275 million in 5th order.

22. Why rotate?
PHYSICS: moment of inertia, use of matrices to rotate vectors in 3D, 4D space-time, etc., …
Loly: Mathematical Gazette – (semi-)magic square theorem.
Rogers and Loly: American Journal of Physics - (semi-)magic cube – inertia tensor theorem.
8 phases – Frank Hruska’s A and B matrices turn out to be one from each of the α and β interlacing sets on rotation and reflection – Styan and Chu’s sweet and sour.

23. Windsor IWMS 2007 – LAA preprint – Loly, Cameron, Trump and Schindel
From the 880 there are a few singular magic squares in Dudeney groups I, II and III which are non-diagonable, 8 in group III are ND in both the α and β interlacing sets, e.g., on rotation of a quarter turn, while 16 in each of groups I and II alternate between ND and diagonable. F109 is an example of the latter, F790 of the former. 600 in groups I to VI are singular but diagonable.

24. Rotation of F109 group 1, pandiagonal8 11 14 15 10 5 4 6 3 16 9 12 13 2 7
Charpoly (Maple):
x(x-34)(x2-64)=0
eigenvalues: 34, ±8, 0
rank 3, DIAGONABLE
SVD: 34, , , 0 12 6 15 1 13 3 10 8 2 16 5 11 7 9 4 14
34, 4√17, 2√ 17, 0
Rotated F109:
Charpoly:
x3(x-34)=0
eigenvalues : 34, 0, 0, 0
NON DIAGONABLE

25. all regular (group 3) ATA’s are bisymmetric (M=JMTJ) Example: F790 regular5 4 16 9 11 14 2 7 10 15 3 6 8 1 13 12
EV: 34,0,0,0
NO CHANGE ON ROTATION
SVD: 34, 8√5, 2√5, 0
EV(ATA ): 1156, 320, 20, 0
F803 has same EV’s
310
332
236
278
438
150
N.B. F803 has a different ATA matrix, but same EVs, SVD
378
212
206
360
370
368

26. Deflating F790 regular 5 4 16 9 11 14 2 7 10 15 3 6 8 1 13 12
EV: 34,0,0,0
NO CHANGE ON ROTATION
SVD: 34, 8√5, 2√5, 0
EV(ATA ): 1156, 320, 20, 0
-3.5
-4.5
7.5
0.5
2.5
5.5
-6.5
-1.5
1.5
6.5
-5.5
-2.5
-0.5
-7.5
4.5
3.5
EV: 0,0,0,0
SVD: 8√5, 2√5, 0,0
Rank 2
On compounding there are no eigenvalues for n=16,64, etc.,