Journl of pplied themtics nd Computtionl echnics 208, 7(), 29-36.mcm.pcz.pl p-issn 2299-9965 DOI: 0.752/jmcm.208..03 e-issn 2353-0588 NOE ON RCES OF RIX PRODUCS INVOLVING INVERSES OF POSIIVE DEFINIE ONES ndrzej Z. Grzyboski Institute of themtics, Czestocho University of echnology Czestocho, Polnd ndrzej.grzyboski@im.pcz.pl Received: December 207; ccepted: 7 Februry 208 bstrct. his short note is devoted to the nlysis of the trce of product of to mtrices in the cse here one of them is the inverse of given positive definite mtrix hile the other is nonnegtive definite. In prticulr, reltion beteen the trce of H nd the vlues of digonl elements of the originl mtrix is nlysed. SC 200: 509, 542, 563 Keyords: mtrix product, trce inequlities, inverse mtrix. Introduction rces of mtrix products re of specil interest nd hve ide rnge of pplictions in different fields of science such s economics, engineering, finnce, hydrology nd physics. hey lso rise nturlly in the pplictions of mthemticl sttistics, especilly in regression nlysis, [-3] or the nlysis of discrete-time sttionry processes [4]. here re ppers devoted to the role of mtrix-product-trces in the description of the probbility distributions of qudrtic forms of rndom vectors, [, 5], or to the development of pproximte boundries for their (i.e. product trces) vlues [6-]. In some sttisticl pplictions the product under considertion involves inverse of given positive definite mtrix. In prticulr, it tkes plce in the Byesin nlysis in regression modelling, here the mtrix cn be interpreted s the covrince mtrix of the disturbnces nd/or priori distribution of unknon system-prmeters [2, 3]. In this pper, e present n eqution concerning trces of certin mtrix products involving n inverse of given mtrix, nd next this eqution is used to obtin result relting the chnges in the vlues of the digonl elements of the originl mtrix ith the vlues of the considered trce. his note is orgnized s follos. In the next section e recll some definitions nd fcts tht ill be necessry to stte nd prove the ne results. In Section 3, e stte the min eqution nd then, in Section 4 e present some possible pplictions.
30.Z. Grzyboski 2. Preliminry definitions, fcts nd nottion ll of the mtrices considered here re rel. For ny squre mtrix [ ij ] n n c the symbol ij denotes the cofctor of the element ij, nd dj() denotes the trnspose of mtrix ith elements being the cofctors of pproprite elements of, i.e. dj() [ ]. c ij For ny squre mtrix e rite > 0 (or 0) if the mtrix is positive definite (or positive semi-definite), i.e. is symmetric nd x x > 0 for ll nonzero column vectors x R n (or x x 0 for ll x R n ). squre mtrix is nonnegtive definite if it is positive definite or positive semi-definite one. he folloing fcts concerning determinnts nd/or inverses of mtrices expressed in so-clled block forms cn be found in vrious textbooks, see e.g. [2, pp. 33-34]. Fct. Let [ ij ] k k nd x be k-dimensionl column vector. hen for ny number det x x det x(dj( ) x ) Fct 2. Let nd B be symmetric mtrices. he folloing equlity is true, provided tht the inverses tht occur in this expression do exist () C C B + DE E D D DE E (2) ith mtrices D nd E being defined s follos: D C nd E B C C. Fct 3. Let be nonsingulr nd let u, v be to column vectors ith dimensions equl to the order of. hen ( ) ( u)( v ) uv + (3) + v u his useful eqution gives method of computing the inverse of the left-hnd side of (3) knoing only the inverse of. Definition (Hdmrd Product). If [ ij ] n m nd B [b ij ] n m re the mtrices of the sme dimensions nxm, then their Hdmrd product is the nxm - mtrix А*В of elementise products, i.e. А*В [ ij b ij ] n m We hve the folloing results involving Hdmrd products. Fct 4. For ny squre mtrices, B of the sme order, the folloing equlity holds: r B e (*B ) e (4)
Note on trces of mtrix products involving inverses of positive definite ones 3 ith e being the vector of n pproprite dimension ith ll coefficients equl to unity, i.e. e (,,..,). Schur s lemm. Let nd B be squre mtrices of the sme order. If these mtrices re both nonnegtive definite then their Hdmrd product А*В is lso nonnegtive definite. 3. he min result Let us consider squre symmetric mtrix of order k given in the folloing block-form (5) Let mtrix C hve the folloing block-form: d C L d d( d L )( ) (6) here the constnt d equls Proposition Let > 0 nd H 0. Let nd H be the submtrices obtined by deleting the first ro nd the first column in nd H, respectively. hen the mtrix C given by (6) is positive definite, the constnt d in (6) is positive one, nd r( H) r( H ) r CH (7) Proof. Let us note tht > 0 nd det( ) > 0 (it is becuse > 0). hus the inverse of cn be computed ith the help of the Fcts, 2. Indeed, let us express the mtrices D, E tht ppers in formul (2) using the objects from the mtrix given in (5): D / nd E / No, by (2), the inverse of the block mtrix tkes on the form
32.Z. Grzyboski + L L (8) It results from the Fct 3 tht e hve the folloing equlity for loer-right block in (8): + (9) king the formul (9) into ccount e cn obtin ne forms for the remining three blocks in (8). he first one, in the upper left corner (s mtter of fct x mtrix), tkes on the folloing form: + + + + + 2 + + 2 + + + he second block in the first ro cn be trnsformed in the folloing y: + + + +
Note on trces of mtrix products involving inverses of positive definite ones 33 Similrly the third one tkes on the form: No the mtrix cn be expressed by the folloing simple formul: d L L + ( )( ) d 0 0 d L L + L L C + D ( )( ) 0 (0) hus, for ny mtrix H of n pproprite dimension the folloing equlities hold: r( H) r(ch + DH) r(ch) + r(dh) r(ch) + r( H ) No e sho tht the number d is positive. Indeed, in vie of Fct e hve: det det (dj( ) ) But dj( det( ), so e finlly hve: ) det( ) det( ) Since the mtrix is positive definite one, both bove determinnts re positive nd this yields the positivity of d: d > 0 o complete the proof e need to sho tht the mtrix C is positive definite one. For n rbitrry k-dimensionl vector x (x,x 2 ), x R, x 2 R k e hve x C x d (x b) 2, ith b x 2.
34.Z. Grzyboski his yields tht for ny nonzero vector x R k, x C x > 0 nd thus the mtrix C is positive definite. he proof of Proposition is completed. 4. Some pplictions First e stte proposition hich is quite strightforrd conclusion from Proposition. Proposition 2 Let the mtrices, H stisfy the ssumptions from Proposition, nd let, s previously indicted, symbols nd H denote the submtrices obtined by deleting the first ro nd the first column in nd H, respectively. hen tr( H) tr( H ) 0 Proof. From Proposition e kno tht r( H) r( H ) r CH nd tht C is positive definite mtrix. From Fct 4 e hve: r(ch) e (C H ) e By the ssumptions mtrix H is nonnegtive definite, thus H is lso nonnegtive definite. From nonnegtive definiteness of the mtrices C nd H it follos, in the light of the Schur s lemm, tht C H is nonnegtive definite s ell, nd consequently e (C H ) e 0. his fct completes the proof. Let us consider no symmetric positive definite mtrix [ ij ]. Let us define mtrix x [α ij ] relted ith by the formul: α f(x) nd α ij ij for ll remining elements of, here f : D R, D R, is given rel function. Proposition 3 Let > 0 nd H 0 be squre mtrices of the sme order. Let f be function differentible on n intervl D such tht x > 0 for ll x D. Let for ll x D, function : D R be defined s (x) r( x H ). hen is nondecresing (non- incresing) if nd only if f is nondecresing (nonincresing). Proof. It follos from Proposition tht (x) r( x H ) r( H ) + D(x)r CH/d
Note on trces of mtrix products involving inverses of positive definite ones 35 here D( x) f ( x) While the mtrix C nd constnt d re defined in (6). Note tht the function depends on its rgument only thru the function D. No little clcultion shos tht for ech x Int(D) the derivtive of does exist nd cn be expressed in the folloing form: ' ( x) f D( x) rch '( x) d 2 In vie of the ssumptions bout the mtrices nd H nd our previous results, 2 D( x) rch the rtio is nonnegtive, hich completes the proof. d 5. Conclusions Due the fct tht in our results the mtrix H is nonnegtive definite, one my consider mtrices of the form H, ith being column vector. Becuse of the ell-knon reltion r( ), it is esy to see tht the bove results cn be lso used for the nlysis of the qudrtic forms ith being given symmetric positive definite mtrix. References [] Bo, Y., & Ullh,. (200). Expecttion of qudrtic forms in norml nd nonnorml vribles ith pplictions. Journl of Sttisticl Plnning nd Inference, 40(5), 93-205. [2] Berger, J.O. (985). Sttisticl Decision heory nd Byesin nlysis. Ne York: Springer- Verlg. [3] Grzyboski,. (2002). etody ykorzystni informcji priori estymcji prmetró regresji. Seri onogrfie No 89. Częstocho: Wydnicto Politechniki Częstochoskiej, 53 (in Polish). [4] Ginovyn,.S., & Shkyn,.. (203). On the trce pproximtions of products of oeplitz mtrices. Sttistics nd Probbility Letters, 83, 753-760. [5] gnus, J.R. (979). he expecttion of products of qudrtic forms in norml vribles: the prctice. Sttistic Neerlndic, 33, 3-36. [6] Chng, D.-W. (999). mtrix trce inequlity for products of Hermitin mtrices. Journl of themticl nlysis nd pplictions, 237, 72-725. [7] Fng, Y., Lopro, K.., & Feng, X. (994). Inequlities for the trce of mtrix product. IEEE rnsctions On utomtic Control, 39, 2. [8] Furuichi, S., Kuriym, K., & Yngi, K. (2009). rce inequlities for products of mtrices. Liner lgebr nd its pplictions, 430, 227-2276.
36.Z. Grzyboski [9] Komroff, N. (2008). Enhncements to the von Neumnn trce inequlity. Liner lgebr nd its pplictions, 428, 738-74. [0] Ptel, R., & od,. (979). rce Inequlities Involving Hermitin trices. Liner lgebr nd its pplictions, 23, 3-20. [] Wng, B.-Y., Xi, B.-Y., & Zhng, F. (999). Some inequlities for sum nd product of positive semidefinite mtrices. Liner lgebr nd its pplictions, 293, 39-49. [2] Ro, C.R. (973). Liner Sttisticl Inference nd Its pplictions. Ne York: John Wiley & Sons.