References • R.S. Wikramaratna, The centro-invertible matrix: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 identity matrix – 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-invertible matrix – 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 integer 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 determinant 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-diagonal matrix 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 minor 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 field, 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 matrix congruence X2 = I (mod pa), Amer. integers, 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 integer matrix X, the number of centro-invertible a a+1 M 70 e matrices modulo p for an odd prime p and aN0 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 4.