Four-dimensional Vector Matrix Determinant and Inverse

Transcription

1 I.J. Egieerig ad Maufacturig Published Olie Jue 01 i MECS ( DOI: /iem vailable olie at Four-dimesioal Vector Matrix Determiat ad Iverse J Sag ad H J Bao H X Che G Q Zhao School of Commuicatio Egieerig Jili Uiversity Chagchu 1300 Chia bstract he theory of two-dimesioal matrix has bee popularized i multi-dimesioal matrix. However applicatios of multi-dimesioal matrix also brig space redudacy ad time redudacy we put forward a multidimesioal vector matrix model. his is ew series of study to defie multidimesioal vector matrix mathematics icludig four-dimesioal vector matrix determiat four-dimesioal vector matrix iverse ad related properties. here is iovative cocept of multi-dimesioal vector matrix mathematics created by author with umerous applicatios i egieerig math video coferecig 3D V ad other fields. Idex erms: Multidimesioal vector matrix four-dimesioal vector matrix determiat four-dimesioal vector matrix iverse 01 Published by MECS Publisher. Selectio ad/or peer review uder resposibility of the Research ssociatio of Moder Educatio ad Computer Sciece. 1. Itroductio his paper brigs a ew brach of mathematics called multidimesioal vector matrix mathematics ad its ew subsets four-dimesioal vector matrix determiat ad four-dimesioal vector matrix iverse. he traditioal matrix mathematics [1] that egieerig sciece ad math studets are usually itroduced to i college hadles matrices of oe or two dimesios. shu M. G. Solo [] also defied some multidimesioal matrix algebra operatios. Based o these theories ad papers multidimesioal vector matrix exteds traditioal matrix math to ay figure of dimesios. herefore traditioal matrix math is a subset of multidimesioal vector matrix mathematics. his paper maily brigs forward the defiitio of four-dimesioal vector matrix determiat ad the fourdimesioal vector matrix iverse. We adopt the form that is differet from the defiitio of two-dimesioal matrix. But the properties of two-dimesioal matrix determiat ad iverse ca be exteded to the fourdimesioal vector matrix. he extesio of classical matrix mathematics to ay figure of dimesios has various applicatios i may braches of egieerig math image compressio codig ad other fields. We * Correspodig author. address: baoh09@mails.lu.edu.c saga@lu.edu.c

2 Four-dimesioal Vector Matrix Determiat ad Iverse 31 should promote the other applicatios of multidimesioal vector matrix math that could ot be doe without his multidimesioal vector matrix math. Our group has proposed the defiitio of multidimesioal vector matrix multiplicatio of multi-dimesioal vector matrices multidimesioal Walsh orthogoal trasform ad traditioal discrete cosie trasform [3]. heir applicatio i color image compressio ad codig is more ad more commo ad widespread. For oe thig it coquers the restrictio of traditioal two-dimesioal matrix multiplicatio. For aother thig it carries o high efficiecy of traditioal matrix trasform i the aspect of removig redudacy of color space. By meas of multi-dimesioal vector matrix model color image data ca be expressed ad processed i a uified mathematical model ad better compressio results are received. I Sectio a multi-dimesioal vector matrix model will be itroduced ad the related properties will be discussed. I Sectio 3 we will propose the defiitios of four-dimesioal vector matrix determiat ad iverse. Verificatio the truth of formula with regard to the four-dimesioal vector matrix determiat ad iverse will be also give i the same Sectio. Sectio 4 cocludes this paper.. Proposed heory Based o the multidimesioal vector matrix defiitio proposed by our group we will further study fourdimesioal vector matrix adoi matrix determiat iverse matrix ad related properties.. he defiitio of multi-dimesioal vector matrix: a ii array of umbers 1 i two directios (oe directio has M etries ad the other directio has N etries) is called two-dimesioal matrix ad the set of all such matrices is represeted as M M N. array of umbers a i 1 i i i directios (each directio has I i etries 1 i. I i ca be called the order i this directio) is MI called multi-dimesioal matrix ad the set of all such matrices is deoted as 1I I MK K K r If the dimesios of multi-dimesioal matrix 1 M I I J J deoted as 1 m 1 I I1 I... I m M I I J J where m r. 1 m 1 J J1 J... J M. I 1 I m J 1 J [4]. are separated ito two sets ad the matrix is ca be deoted as M IJ where I ad J are for the vectors ca be called multi-dimesioal vector matrix separated accordig to the vector I ad J multidimesioal vector matrix i short[4]. multi-dimesioal matrix has various relevat multi-dimesioal vector matrices whereas a multidimesioal vector matrix has uique relevat multi-dimesioal matrix. B. Multi-dimesioal vector idetity matrix: Let IJ I I... be a multidimesioal vector matrix where 1I I m If vector I J J J J J the IJ is called multidimesioal vector square matrix [5]. 1 i i 0 i Let where i I I... represets vector 1I I m J J... 1 J J if it has the same dimesio the meaigs of i is that m i ad im. If IJ = i IJ E is said to be multi-dimesioal vector idetity matrix deoted as II or E simply. C. Multiplicatio of multi-dimesioal vector matrices:

3 3 Four-dimesioal Vector Matrix Determiat ad Iverse Let IJ ad BUV I I... be two multi-dimesioal vector matrices i which 1I I m U U1 U... U s V V1 V... Vt If J U the IJ ad BUV are multiplicative. J J1 J... J Let IL be I L matrix ad B LJ be L J matrix. he result of multiplicatio of IL ad B LJ is defied C c i as a I J matrix [4] 1 m 1 L1 il l i1 im 1 L l i1 im l1 lk l1 lk 1 1 lk Lk c a b a b Which is deoted as C IJ = IL B LJ. 1 k l1 lk For simplicity the sigal specified this kid of form is default. L L is rewritte as D. Multi-dimesioal vector matrix traspose he defiitio of multi-dimesioal vector matrix traspose L ad the sigal 1 m 1 IJ JI 3. Four-dimesioal Vector Matrix Determiat d Iverse a i i l l k (1) a il is rewritte as.if o he multidimesioal vector matrix determiat for a oe-dimesioal matrix is udefied. he multidimesioal vector matrix determiat for a two-dimesioal square matrix is calculated usig the traditioal methods. he multidimesioal vector matrix determiat of a two-dimesioal o-square matrix is udefied. Hece at first a four-dimesioal vector matrix which ca be calculated determiat should be a fourdimesioal vector square matrix. Secodly commutative matrices must be square matrices with the same orders. m m aii 11 For istats a four-dimesioal vector square matrix mm icludig 1 i1 m 1 i 1 1 m ad 1. For a four-dimesioal square vector matrix m m all the elemets of four vector directios where the a ii elemet i the matrix m m is located ca be cacelled. he other elemets are regularly collected i a matrix with the orders of m1 ad the its determiat ca be calculated. he matrix determiat ca be called the cofactor of the elemet deoted as the ii 11 a ii i 1 i 1 M ii i Mi1i 1 ca be said the vector cofactor of the elemet aii 11. he defiitio of four-dimesioal vector square matrix determiat For a four-dimesioal vector square matrix each elemet of ay vector directio is multiplied by its vector cofactor ad the all the products are added. he product ca be called the four-dimesioal vector square matrix determiat..

4 If =m Four-dimesioal Vector Matrix Determiat ad Iverse 33 m m m... m aii 1 i1 i1 i 1... m i m m m aii 1 i1 1 i 1... or or m m m m m m aii 1 i1 i1 i m m mm mm aii 1 i m 1... m i m i 1... m Similarly all elemets of ay vector directio i the four-dimesioal vector square matrix are multiplied by the vector cofactor of correspodig elemets i aother vector directio ad the all the products are added. he result is zero. If =m I coclusio a i i 1 a i1i 1 a i1i m 1 m a 11i i 1 a i1i 1 a mi1i m 1 a i i 1 a i1i m 1 m a i1i mm 1 mm a 11i i 1 a mi1i m 1 a mmi1i mm 1 i1 1 i... ai 1i ai 1i m1 m 0 i1 1 i i1 1 i... a11i 1 i 11 a 1 mi 1 i m 1 0 i1 1 i () (3) B. he defiitio of four-dimesioal vector square matrix iverse he multidimesioal vector matrix iverse for a oe-dimesioal matrix is udefied. he multidimesioal vector matrix iverse of a two-dimesioal matrix exists if it is a square matrix ad has a ozero determiat ad is calculated usig the stadard meas i traditioal matrix math. For the four-dimesioal vector matrix each four-dimesioal vector square matrix with a ozero determiat is ecessary. Firstly we defie the fourdimesioal vector adoi matrix. he defiitio of four-dimesioal vector adoi matrix m m ii 11 m m If a four-dimesioal vector square matrix mm is ivertible ad m m the m m m m m m If =m m mm m m m m m m m m m C. he properties of four-dimesioal vector square matrix determiat ad iverse (4)

5 34 Four-dimesioal Vector Matrix Determiat ad Iverse I traditioal matrix mathematics if a matrix possesses a iverse ad that matrix is multiplied by its iverse the product is a idetity matrix with the same dimesios. Because multidimesioal vector matrices are a cocateatio of two-dimesioal matrices if a fourdimesioal vector matrix has a iverse ad that four-dimesioal vector matrix is multiplied by its iverse the the product will be a four-dimesioal vector idetity matrix with the same dimesios. Due to the defiitio of four-dimesioal vector matrix iverse ( 4 ) ad matrix m m is the four-dimesioal vector adoi matrix we ca coclude 1 1 m m m m m m m m m m m m 1m1m m m m11 m1 m111 m11 m m1 m m m1 m1 m1 m m m m m 0 if this a 111 a a1111 a 1 11 am 11 am1 am 111 am11 a 11m a 1 m a 11m1 a 1 m1 am 1m am m am 1m1 am m1 m m 0 m m m m UNI m m m m m m 0 m m Due to the multiplicatio of multi-dimesioal vector matrices (1) ad the formula of four-dimesioal vector matrix determiat () ad (3) that is 1 m m UNI mm For example the four-dimesioal vector matrix with two orders is give. By meas of the program s operatio we ca calculate the four-dimesioal vector iverse matrix.

6 Four-dimesioal Vector Matrix Determiat ad Iverse UNI I traditioal matrix mathematics if a matrix is a idetity matrix the determiat of two-dimesioal matrix is 1. Similarly multidimesioal vector matrices are a cocateatio of two-dimesioal matrices if a fourdimesioal vector matrix is a four-dimesioal vector idetity matrix the result of four-dimesioal vector idetity matrix determiat is 1. hat is m m 1. For example a111 a a1111 a1 11 am 11 am1 am 111 am11 m m a11m a1 m a11m1 a1 m1 am 1m am m am 1m1 am m1 m m m m For a four-dimesioal vector square matrix m m m m m aii 1 i1 i1 i a a am11m11 m aii 1 ii 1 1 m a a a11m 11m a 1 ii 1 m m 1 ii

7 36 Four-dimesioal Vector Matrix Determiat ad Iverse So m m =. m a a am11m11 a 1 ii 1 1 ii 1 i11i a a a11m 11 m m m here are still may properties of two-dimesioal matrix that ca be exted to the four-dimesioal vector matrix. If all the elemets of ay vector directio are zero i a four-dimesioal vector square matrix m m the m m = 0. If oe vector directio is proportioal to aother vector directio of a four-dimesioal vector square matrix m m the m m = 0. If oe vector directio is a liear combiatio of oe or more other vector directios of a four-dimesioal vector square matrix m m the m m = 0. If two vector directios of a four-dimesioal vector square matrix m m are iterchaged the sig of the determiat of the matrix m m is chaged. four-dimesioal vector square matrix m m iverse which it is a ivertible matrix ca be uique. If four-dimesioal vector square matrix mm m m m m is ivertible. 1 1 If four-dimesioal vector square matrix mm is ivertible 0 ad m m is also ivertible the m m m m If four-dimesioal vector square matrix m m ad If four-dimesioal vector square matrix are both ivertible ad fourdimesioal vector square matrix m m 1-1 m m m m m m B m m m m ad B m m is also ivertible the are both ivertible the m m Bm m Bm m m m here are various properties of four-dimesioal vector matrix determiat ad iverse to prove the correctess of four-dimesioal vector matrix determiat ad iverse defiitio i this paper. Meawhile we ru successfully the correspodig program to verify the defiitio of the four-dimesioal vector matrix determiat ad iverse.

8 Four-dimesioal Vector Matrix Determiat ad Iverse Coclusio O the basis of ewly operatio laws of multidimesioal vector matrix we defie the four-dimesioal vector matrix determiat iverse ad related properties. We also prove the correctess of these formulas by mathematics ad program. I this paper we have itroduced maily the theory of multi-dimesioal vector matrix the four-dimesioal vector matrix determiat ad iverse. he future work is to exted the four-dimesioal vector matrix iverse to multidimesioal vector matrix iverse. We will apply these theories ad defiitios o multidimesioal vector matrix. ckowledgmet his work is sposored by the Natioal Sciece Foudatio of Chia uder Grat ad Refereces [1] Frakli Joel L. [000] Matrix heory. Mieola N.Y.: Dover. [] shu M.G. Solos. Multidimesioal matrix mathematics: multidimesioal matrix traspose symmetry at symmetry determiat ad iverse part 4 of 6. Proceedigs of the World Cogress o Egieerig 010 vol.3 WEC 010 Jue 30-July 010 Lodo U.K. [3] hmed N. Nataraa. ad Rao K. R. O image processig ad a discrete cosie trasform. IEEE ras. Compute l [4] J Sag M S Che H X Che L L Liu ad N Su. Multi-dimesioal vector matrix theory ad its applicatio i color image codig. he Imagig Sciece Joural vol.58 o.3 Jue 010 pp (6). [5] Liu L.L Che H.X Sag.J Su.N. 4D order-4 vector matrix DC iteger trasform ad its applicatio i video codec Imagig Sciece Joural the vol. 58 o. 6 December 010 pp (10).