References

• R.S. Wikramaratna, The centro-invertible :a new type of matrix arising in pseudo-random number generation, Centro-invertible Matrices Linear Algebra and its Applications, 434 (2011) pp144-151. [doi:10.1016/j.laa.2010.08.011]. Roy S Wikramaratna, RPS Energy [email protected] • R.S. Wikramaratna, Theoretical and empirical convergence results for additive congruential random number generators, Reading University (Conference in honour of J. Comput. Appl. Math., 233 (2010) 2302-2311. Nancy Nichols' 70th birthday ) [doi: 10.1016/j.cam.2009.10.015]. 2-3 July 2012

Career Background Some definitions …

• Worked at Institute of Hydrology, 1977-1984 • I is the k by k – Groundwater modelling research and consultancy • J is the k by k matrix with ones on anti-diagonal and zeroes – P/t MSc at Reading 1980-82 (Numerical Solution of PDEs) elsewhere • Worked at Winfrith, Dorset since 1984 – Pre-multiplication by J turns a matrix ‘upside down’, reversing order of terms in each column – UKAEA (1984 – 1995), AEA Technology (1995 – 2002), ECL Technology (2002 – 2005) and RPS Energy (2005 onwards) – Post-multiplication by J reverses order of terms in each row – Oil reservoir engineering, porous medium flow simulation and 0 0 0 1 simulator development   0 0 1 0 – Consultancy to Oil Industry and to Government J = = ()j 0 1 0 0 pq   • Personal research interests in development and   application of numerical methods to solve engineering 1 0 0 0 j =1 if p + q = k +1 problems, and in mathematical and numerical analysis pq

of those methods j pq = 0otherwise

Outline of today’s talk … more definitions …

• Definitions • The matrix centro-rotation operation on a matrix X (to R – Matrix centro-rotation operation be denoted X ) is defined as – Centro- – A component-wise 180 degree rotation about centre of matrix – Equivalent to reversing order of both rows and columns • Theory – Properties XR = JXJ  x x x x  – Examples  44 43 42 41   x34 x33 x32 x31  – Relationship with involutory matrices XR = = ()j  x x x x  ()()k +1− p k+1−q  24 23 22 21  – How common are centro-invertible matrices?    x14 x13 x12 x11 

1 … yet more definitions Simple examples

•A matrix X is defined to be centro-invertible if and only • The matrices I, J defined previously are centro- if X-1=XR invertible, as are –I and –J – All of these matrices are also involutory matrices •A matrix X with components is defined to be – All of these matrices are also centro-symmetric centro-invertible modulo M if and only if the equation – Can show that any two of {centro-invertible, involutory, -1 R X =X holds true when working in modular arithmetic, centro-symmetric} also imply the third modulo (a large integer) M

Observations … Non-trivial examples (modulo 16)

1 13 3  • Centro-invertible matrices (modulo 2k) arise naturally in a 15 2   2 by 2 example   3 by3 example 3 8 6  real application 14 3   ACORN matrix, modulo 16 ACORN matrix, modulo 16 6 1 10 – Additive Congruential Random Number (ACORN) Generator – see reference in JCAM, 2010 1 11 10 6 5  15 5 6 10 11       5 8 13 8 15  11 8 3 8 1  • Choice of name is by analogy with centro-symmetric 5by5examples ()i 15 10 14 7 3  (ii )  1 6 2 9 13      3 0 8 0 6 13 0 8 0 10 matrices (which are invariant under the centro-rotation ACORN matrix, modulo 16   (-I) times ACORN matrix,       operation)  6 5 12 12 14 modulo 16  6 5 12 12 14 – It appears that centro-invertible matrices have not arisen or been described previously in the literature prior to the 2010 0 0 0 15 2  0 0 1 13 3  JCAM reference     0 0 0 14 3  0 0 3 8 6  – The term centro-invertible was first used in the 2011 LAA ()iii  1 13 3 0 0  ()iv  0 0 6 1 10     reference 3 8 6 0 0 15 2 0 0 0   Block anti-diagonal with   Block anti-diagonal with     centro-invertible blocks  6 1 10 0 0  centro-invertible blocks 14 3 0 0 0 

Some properties of centro-invertible … more observations matrices

• There is a key relationship that exists between centro- • The inverse of a centro-invertible matrix is itself centro- invertible matrices and involutory matrices (an invertible involutory matrix Y is one for which Y2=I) •If X is centro-invertible modulo M, then so is X raised to any integer power – Can define a one-one onto mapping between centro- • The of a centro-invertible matrix is equal to +1 invertible matrices and involutory matrices (in fact, there or -1 (for a centro-invertible matrix modulo M the are several such mappings that are possible; in determinant is 1 or M-1) particular any centro invertible matrix is an upside down – Note all ACORN matrices have determinant 1, so there are involutory matrix and vice-verca) centro-invertible matrices that are not ACORN matrices • Any block anti- having centro-invertible sub- blocks on the anti-diagonal and zeroes elsewhere is itself • Allows existing results concerning the number of k by centro-invertible k involutory matrices modulo M to be translated to – Note typographic error in LAA, 2011 – text says ‘involutory’ give analogous results for centro-invertible matrices instead of ‘centro-invertible’ in this statement modulo M

2 Corollary 3 Results … (modulus a power of 2) • The number T(k,2n) of k by k centro-invertible matrices modulo n • THEOREM 2 is given by one of the following equations depending on the value of n, where as before g =1 and g is as in Corollary 2 – There exists a 1 to 1 correspondence between k by k centro- 0 t k / 2  2−t(2k −3t )  invertible matrices and k by k involutory matrices 1   T(k,2 ) = g k ∑   •PROOF t=0  gt g k −2t  – Can show that X is centro-invertible if and only if JX is 2 2 involutory k / 2  2k −4tk +5t  T(k,22 ) = g   – Note, also, X is centro-invertible if and only if XJ is involutory k ∑   t =0  gt gk −2t 

k min(t,k−t) r(3r−2k) 2  2  • Corollary 1 n k 2t(k−t)(n−3)   T(k,2 ) = 2 gk ∑∑2   n ≥ 3 – Number of k by k centro-invertible matrices (modulo M) for any t=0 r=0  gr gt−r gk −t−r  k is identical to the number of k by k involutory matrices • PROOF (modulo M) – By analogy with Hodges (1958) and Levine and Korfhage (1964); making use of Corollary 1. [Note typo in L&K for the case n=2, corrected above].

n1 n2 nr M = p1 p2 ...pr

References for Corollaries 2, 3 and 4 Corollary 4 (general result, any (involutory matrices) modulus)

• Let p1, p2, …, pr be distinct primes and let the prime power factorisation of M be • J.H. Hodges, The matrix equation X2-I=0 over a finite , n1 n2 nr Amer. Math. Monthly, 65 (1958), pp. 518-520 M = p1 p2 ...pr • Then the number of k by k centro-invertible matrices over the • I. Reiner, The X2 = I (mod pa), Amer. , modulo M is Math. Monthly, 67 (1960), pp.773-775. r n j T(k, M ) = ∏T (k, p j ) j=1 • J. Levine and R.R. Korfhage, Automorphisms of abelian

nj groups induced by involutory matrices, general modulus, where T(k,pj ) is as defined in corollary 2 for odd values of pj Duke Math. J., 31 (1964), pp.631-653. and corollary 3 if pj is equal to 2

• PROOF – By analogy with Levine and Korfhage (1964); making use of Corollary 1.

Corollary 2 100 (prime-power modulus, odd primes) 90 )

s 80 e c i r t

•For a k by k X, the number of centro-invertible a

a+1 M 70 e

matrices modulo p for an odd prime p and a≥0 is given by l b i t

r 10 by 10 e

v 60 n I - o

r 9 by 9 k t n a+1  g 2t ()k−t a  e 50 k C

T(k, p ) =  ⋅ p  k 8 by 8 ∑  y g g b t=0  t k −t  k 40 f

o 7 by 7 r e b 30 m

u 6 by 6

where g =1 and g is given by N

0 t ( 0 1 g

t t−1 o 20 2 l 5 by 5 g = pt 1− p−i = pt − pi 0 < t ≤ k t ∏()()∏ 4 by 4 i=1 i=0 10 3 by 3 2 by 2 • PROOF 0 – By analogy with Reiner (1960); making use of Corollary 1. 1 10 100 Modulus M

Figure 1: Number of k by k centro-invertible matrices (mod M), plotted against the modulus M on logarithmic scales, base 10

3 Modulus M 1 10 100 0 2 by 2

3 by 3 -10 )

e 4 by 4 l b i t r

e -20 5 by 5 v n I - o r t 6 by 6 n -30 e C e r

a 7 by 7 t -40 a h t s e

c 8 by 8 i

r -50 t a M k

y -60 9 by 9 b k f o

n 10 by 10

o -70 i t r o p o r

P -80 ( 0 1 g o l -90

-100

Figure 2: Proportion of k by k matrices (mod M) that are centro-invertible, plotted against modulus M on logarithmic scales, base 10

Observations

• Results suggest an approximate relationship between the number of centro-invertible k by k matrices (or equivalently the number of k by k involutory matrices) that exist modulo M and the total number (N=M(kxk)) of possible k by k matrices modulo M as follows – Number of centro-invertible matrices ~ N 0.5=M (kxk)/2 – Proportion that are centro-invertible ~ N -0.5=M –(kxk)/2

• NOTE Purely empirical relationship at present

• QUESTION Is it possible to infer these (or similar) expressions from Corollaries 2 – 4 based on theoretical analysis?

… thank you for listening

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