Generalised matrix multivariate Pearson type II-distribution
aa r X i v : . [ m a t h . S T ] J un Generalised matrix multivariate Pearson typeII-distribution
Jos´e A. D´ıaz-Garc´ıa ∗ Universidad Aut´onoma Agraria Antonio Narro
Ram´on Guti´errez-S´anchez † University of Granada
Abstract
Matrix multivariate Pearson type II-Riesz distribution is definedand some of its properties are studied. In particular, the associated matrixmultivariate beta distribution type I is derived. Also the singular values andeigenvalues distributions are obtained.
Keywords
Matrix multivariate; Pearson Type II distribution; Riesz dis-tribution; Kotz-Riesz distribution; real, complex,quaternion and octonionrandom matrices; real normed division algebras.
When a new statistic theory is proposed, the statistician known well about therigourously mathematical foundations of their discipline, however in order to reacha wider interdisciplinary public, some of the classical statistical techniques havebeen usually published without explaining the supporting abstract mathematicaltools which governs the approach. For example, in the context of the distributiontheory of random matrices, in the last 20 years, a number of more abstract andmathematical approaches have emerged for studying and generalising the usual ma-trix variate distributions. In particular, this needing have appeared recently in the ∗ Corresponding author: Jos´e A. D´ıaz-Garc´ıa, Universidad Aut´onoma Agraria Antonio Narro,Calzada Antonio Narro 1923, Col. Buenavista, 25315 Saltillo, Coahuila, M´exico, E-mail: [email protected] † Department of Statistics and O.R, University of Granada, Granada 18071, Spain, E-mail:[email protected] et al. [26], Ratnarajah et al. [27], Edelman and Rao [12], Forrester [15], among manyothers authors. Studying distribution theory by another algebras, beyond real, haveled several generalisations of substantial understanding in the theoretical context,and we expect that it is more extensively applied when a an improvement of itsunified potential can be explored in other contexts. Two main tendencies have beenconsidered in literature, Jordan algebras and real normed division algebras. Someworks dealing the first approach are due to Faraut and Kor´anyi [14], Massam [24],Casalis, and Letac [3], Hassairi and Lajmi [19], Hassairi et al. [20, 21], Ko lodziejek[23], and the references therein, meanwhile, the second technique has been studiedby Gross and Richards [16], D´ıaz-Garc´ıa and Guti´errez-J´aimez [7], D´ıaz-Garc´ıa[4], D´ıaz-Garc´ıa [5, 6], among many others.In the same manner, different generalisations of the multivariate statistical anal-ysis have been proposed recently. This generalised technique studies the effect ofchanging the usual matrix multivariate normal support by a general matrix mul-tivariate family of distributions, such as the elliptical contoured distributions (orsimply, matrix multivariate elliptical distributions), see Fang and Zhang [13] andGupta and Varga [18]. This family of distributions involves a number of knownmatrix multivariate distributions such as normal, Kotz type, Bessel, Pearson typeII and VII, contaminated normal and power exponential, among many others. Twoimportant properties of these distributions must be emphasise: i). Matrix multi-variate elliptical distributions provide more flexibility in the statistical modeling byincluding distributions with heavier or lighter tails and/or greater or lower degreeof kurtosis than matrix multivariate normal distribution; and, ii). Most of the sta-tistical tests based on matrix multivariate normal distribution are invariant underthe complete family of matrix multivariate elliptical distributions.Recently, a slight combination of these two theoretical generalisations have ap-peared in literature; namely, Jordan algebras has been led to the matrix multivariateRiesz distribution and its associated beta distribution. D´ıaz-Garc´ıa [6] proved thatthe above mentioned distributions can be derived from a particular matrix multi-variate elliptical distribution, termed matrix multivariate Kotz-Riesz distribution.Similarly, matrix multivariate Riesz distribution is also of interest from the math-ematical point of view; in fact most of their basic properties under the structuretheory of normal j -algebras and the theory of Vinberg algebras in place of Jordan2lgebras have been studied by Ishi [22] and Boutouria and Hassiri [2], respectively.In this scenario, we can now propose a generalisation of the matrix multivariatebeta, T and Pearson type II distributions based on a matrix multivariate Kotz-Rieszdistribution. As usual in the normal case, extensions of beta, T and Pearson type IIdistributions involves two alternatives, the matricvariate and the matrix multivariateversions , see D´ıaz-Garc´ıa [4], D´ıaz-Garc´ıa [5, 6], D´ıaz-Garc´ıa and Guti´errez-J´aimez[7, 8, 9] and D´ıaz-Garc´ıa and Guti´errez-S´anchez [10].This article derives the matrix multivariate beta and Pearson type II distribu-tions obtained from a matrix multivariate Kots-Riesz distribution and some of theirbasic properties are studied. Section 2 gives some basic concepts and the notation ofabstract algebra, Jacobians and distribution theory. The nonsingular central matrixmultivariate Pearson type II-Riesz distribution and the corresponding generalisedmatrix multivariate beta type I distribution are studied in Section 3. Finally, thejoint densities of the singular values are derived in Section 4. A detailed discussion of real normed division algebras can be found in Baez [1] andNeukirch et al. [25]. For your convenience, we shall introduce some notation,although in general, we adhere to standard notation forms.For our purposes: Let F be a field. An algebra A over F is a pair ( A ; m ), where A is a finite-dimensional vector space over F and multiplication m : A × A → A isan F -bilinear map; that is, for all λ ∈ F , x, y, z ∈ A , m ( x, λy + z ) = λm ( x ; y ) + m ( x ; z ) m ( λx + y ; z ) = λm ( x ; z ) + m ( y ; z ) . Two algebras ( A ; m ) and ( E ; n ) over F are said to be isomorphic if there is aninvertible map φ : A → E such that for all x, y ∈ A , φ ( m ( x, y )) = n ( φ ( x ) , φ ( y )) . By simplicity, we write m ( x ; y ) = xy for all x, y ∈ A . The term matricvariate distribution was first introduced Dickey [11], but the expression matrix-variate distribution or matrix variate distribution or matrix multivariate distribution was later usedto describe any distribution of a random matrix, see Gupta and Nagar [17] and Gupta and Varga[18], and the references therein. When the density function of a random matrix is written includingthe trace operator then the matrix multivariate designation shall be used. A be an algebra over F . Then A is said to be1. alternative if x ( xy ) = ( xx ) y and x ( yy ) = ( xy ) y for all x, y ∈ A ,2. associative if x ( yz ) = ( xy ) z for all x, y, z ∈ A ,3. commutative if xy = yx for all x, y ∈ A , and4. unital if there is a 1 ∈ A such that x x = 1 x for all x ∈ A .If A is unital, then the identity 1 is uniquely determined.An algebra A over F is said to be a division algebra if A is nonzero and xy =0 A ⇒ x = 0 A or y = 0 A for all x, y ∈ A .The term “division algebra”, comes from the following proposition, which showsthat, in such an algebra, left and right division can be unambiguously performed.Let A be an algebra over F . Then A is a division algebra if, and only if, A isnonzero and for all a, b ∈ A , with b = 0 A , the equations bx = a and yb = a haveunique solutions x, y ∈ A .In the sequel we assume F = ℜ and consider classes of division algebras over ℜ or “ real division algebras ” for short.We introduce the algebras of real numbers ℜ , complex numbers C , quaternions H and octonions O . Then, if A is an alternative real division algebra, then A isisomorphic to ℜ , C , H or O .Let A be a real division algebra with identity 1. Then A is said to be normed ifthere is an inner product ( · , · ) on A such that( xy, xy ) = ( x, x )( y, y ) for all x, y ∈ A . If A is a real normed division algebra , then A is isomorphic ℜ , C , H or O .There are exactly four normed division algebras: real numbers ( ℜ ), complexnumbers ( C ), quaternions ( H ) and octonions ( O ), see Baez [1]. We take into accountthat should be taken into account, ℜ , C , H and O are the only normed divisionalgebras; furthermore, they are the only alternative division algebras.Let A be a division algebra over the real numbers. Then A has dimension either1, 2, 4 or 8. Finally, observe that ℜ is a real commutative associative normed division algebras, C is a commutative associative normed division algebras, H is an associative normed division algebras, O is an alternative normed division algebras.4et L βn,m be the set of all n × m matrices of rank m ≤ n over A with m distinctpositive singular values, where A denotes a real finite-dimensional normed divisionalgebra . Let A n × m be the set of all n × m matrices over A . The dimension of A n × m over ℜ is βmn . Let A ∈ A n × m , then A ∗ = ¯ A T denotes the usual conjugatetranspose.Table 1 sets out the equivalence between the same concepts in the four normeddivision algebras. Table 1: NotationReal Complex Quaternion Octonion GenericnotationSemi-orthogonal Semi-unitary Semi-symplectic Semi-exceptionaltype V βm,n Orthogonal Unitary Symplectic Exceptionaltype U β ( m )Symmetric Hermitian Quaternionhermitian Octonionhermitian S βm We denote by S βm the real vector space of all S ∈ A m × m such that S = S ∗ . Inaddition, let P βm be the cone of positive definite matrices S ∈ A m × m . Thus, P βm consist of all matrices S = X ∗ X , with X ∈ L βn,m ; then P βm is an open subset of S βm .Let D βm consisting of all D ∈ A m × m , D = diag( d , . . . , d m ). Let T βU ( m ) bethe subgroup of all upper triangular matrices T ∈ A m × m such that t ij = 0 for1 < i < j ≤ m . Let Z ∈ L βn,m , define the norm of Z as || Z || = √ tr Z ∗ Z .For any matrix X ∈ A n × m , d X denotes the matrix of differentials ( dx ij ). Finally,we define the measure or volume element ( d X ) when X ∈ A n × m , S βm , D βm or V βm,n ,see D´ıaz-Garc´ıa and Guti´errez-J´aimez [7] and D´ıaz-Garc´ıa and Guti´errez-J´aimez [9].If X ∈ A n × m then ( d X ) (the Lebesgue measure in A n × m ) denotes the exteriorproduct of the βmn functionally independent variables( d X ) = n ^ i =1 m ^ j =1 dx ij where dx ij = β ^ k =1 dx ( k ) ij . If S ∈ S βm (or S ∈ T βU ( m ) with t ii > i = 1 , . . . , m ) then ( d S ) (the Lebesguemeasure in S βm or in T βU ( m )) denotes the exterior product of the exterior productof the m ( m − β/ m functionally independent variables,( d S ) = m ^ i =1 ds ii m ^ i>j β ^ k =1 ds ( k ) ij . d S ) defined thus, it is required that S ∈ P βm ,that is, S must be a non singular Hermitian matrix (Hermitian definite positivematrix).If Λ ∈ D βm then ( d Λ ) (the Legesgue measure in D βm ) denotes the exterior productof the βm functionally independent variables( d Λ ) = n ^ i =1 β ^ k =1 dλ ( k ) i . If H ∈ V βm,n then ( H ∗ d H ) = m ^ i =1 n ^ j = i +1 h ∗ j d h i . where H = ( H ∗ | H ∗ ) ∗ = ( h , . . . , h m | h m +1 , . . . , h n ) ∗ ∈ U β ( n ). It can be proved thatthis differential form does not depend on the choice of the H matrix. When n = 1; V βm, defines the unit sphere in A m . This is, of course, an ( m − β - dimensionalsurface in A m . When n = m and denoting H by H , ( H d H ∗ ) is termed the Haarmeasure on U β ( m ).The surface area or volume of the Stiefel manifold V βm,n isVol( V βm,n ) = Z H ∈V βm,n ( H d H ∗ ) = 2 m π mnβ/ Γ βm [ nβ/ , (1)where Γ βm [ a ] denotes the multivariate Gamma function for the space S βm and isdefined as Γ βm [ a ] = Z A ∈ P βm etr {− A }| A | a − ( m − β/ − ( d A )= π m ( m − β/ m Y i =1 Γ[ a − ( i − β/ , and Re( a ) > ( m − β/
2. This can be obtained as a particular case of the generalisedgamma function of weight κ for the space S βm with κ = ( k , k , . . . , k m ) ∈ ℜ m , taking κ = (0 , , . . . , ∈ ℜ m and which for Re( a ) ≥ ( m − β/ − k m is defined by, seeGross and Richards [16] and Faraut and Kor´anyi [14],Γ βm [ a, κ ] = Z A ∈ P βm etr {− A }| A | a − ( m − β/ − q κ ( A )( d A ) (2)= π m ( m − β/ m Y i =1 Γ[ a + k i − ( i − β/ a ] βκ Γ βm [ a ] , (3)6here etr( · ) = exp(tr( · )), | · | denotes the determinant, and for A ∈ S βm q κ ( A ) = | A m | k m m − Y i =1 | A i | k i − k i +1 (4)with A p = ( a rs ), r, s = 1 , , . . . , p , p = 1 , , . . . , m is termed the highest weight vector ,see Gross and Richards [16], Faraut and Kor´anyi [14] and Hassairi and Lajmi [19];And, [ a ] βκ denotes the generalised Pochhammer symbol of weight κ , defined as[ a ] βκ = m Y i =1 ( a − ( i − β/ k i = π m ( m − β/ m Y i =1 Γ[ a + k i − ( i − β/ βm [ a ]= Γ βm [ a, κ ]Γ βm [ a ] , where Re( a ) > ( m − β/ − k m and( a ) i = a ( a + 1) · · · ( a + i − , is the standard Pochhammer symbol.Additional, note that, if κ = ( p, . . . , p ), then q κ ( A ) = | A | p . In particular if p = 0,then q κ ( A ) = 1. If τ = ( t , t , . . . , t m ), t ≥ t ≥ · · · ≥ t m ≥
0, then q κ + τ ( A ) = q κ ( A ) q τ ( A ), and in particular if τ = ( p, p, . . . , p ), then q κ + τ ( A ) ≡ q κ + p ( A ) = | A | p q κ ( A ). Finally, for B ∈ T βU ( m ) in such a manner that C = B ∗ B ∈ S βm , q κ ( B ∗ AB ) = q κ ( C ) q κ ( A ), and q κ ( B ∗− AB − ) = ( q κ ( C )) − q κ ( A ) = q − κ ( C ) q κ ( A ) , see Hassairi et al. [21].Finally, the following Jacobians involving the β parameter, reflects the gener-alised power of the algebraic technique; the can be seen as extensions of the fullderived and unconnected results in the real, complex or quaternion cases, see Farautand Kor´anyi [14] and D´ıaz-Garc´ıa and Guti´errez-J´aimez [7]. These results are thebase for several matrix and matric variate generalised analysis. Proposition 2.1.
Let X and Y ∈ L βn,m be matrices of functionally independentvariables, and let Y = AXB + C , where A ∈ L βn,n , B ∈ L βm,m and C ∈ L βn,m areconstant matrices. Then ( d Y ) = | A ∗ A | mβ/ | B ∗ B | mnβ/ ( d X ) . (5)7 roposition 2.2 (Singular Value Decomposition, SV D ) . Let X ∈ L βn,m be matrixof functionally independent variables, such that X = W DV ∗ with W ∈ V βm,n , V ∈ U β ( m ) and D = diag( d , · · · , d m ) ∈ D m , d > · · · > d m > . Then ( d X ) = 2 − m π ̺ m Y i =1 d β ( n − m +1) − i m Y i Let X ∈ L βn,m be matrix of functionally independent variables,and write X = V T , where V ∈ V βm,n and T ∈ T βU ( m ) with positive diagonalelements. Define S = X ∗ X ∈ P βm . Then ( d X ) = 2 − m | S | β ( n − m +1) / − ( d S )( V ∗ d V ) . (7)Finally, to define the matrix multivariate Pearson type II-Riesz distribution weneed to recall the following two definitions of Kotz-Riesz and Riesz distributions.From D´ıaz-Garc´ıa [6]. Definition 2.1. Let Σ ∈ Φ βm , Θ ∈ Φ βn , µ ∈ L βn,m and κ = ( k , k , . . . , k m ) ∈ ℜ m .And let Y ∈ L βn,m and U ( B ) ∈ T βU ( n ), such that B = U ( B ) ∗ U ( B ) is the Choleskydecomposition of B ∈ S βm . Then it is said that Y has a Kotz-Riesz distribution oftype I and its density function is β mnβ/ P mi =1 k i Γ βm [ nβ/ π mnβ/ Γ βm [ nβ/ , κ ] | Σ | nβ/ | Θ | mβ/ × etr (cid:8) − β tr (cid:2) Σ − ( Y − µ ) ∗ Θ − ( Y − µ ) (cid:3)(cid:9) × q κ (cid:2) U ( Σ ) ∗− ( Y − µ ) ∗ Θ − ( Y − µ ) U ( Σ ) − (cid:3) ( d Y ) (8)with Re( nβ/ > ( m − β/ − k m ; denoting this fact as Y ∼ KR β,In × m ( κ, µ , Θ , Σ ) . From Hassairi and Lajmi [19] and D´ıaz-Garc´ıa [4].8 efinition 2.2. Let Ξ ∈ Φ βm and κ = ( k , k , . . . , k m ) ∈ ℜ m . Then it is said that V has a Riesz distribution of type I if its density function is β am + P mi =1 k i Γ βm [ a, κ ] | Ξ | a q κ ( Ξ ) etr {− β Ξ − V }| V | a − ( m − β/ − q κ ( V )( d V ) (9)for V ∈ P βm and Re( a ) ≥ ( m − β/ − k m ; denoting this fact as V ∼ R β,Im ( a, κ, Ξ ). A detailed discussion of Riesz distribution may be found in Hassairi and Lajmi [19]and D´ıaz-Garc´ıa [4]. In addition the Kotz-Riesz distribution is studied in detailin D´ıaz-Garc´ıa [6]. For your convenience, we adhere to standard notation stated inD´ıaz-Garc´ıa [4], D´ıaz-Garc´ıa [6]. Theorem 3.1. Let (cid:16) S / (cid:17) = S ∼ R β,I ( νβ/ , k, , k ∈ ℜ and Re( νβ/ > − k ; independent of Y ∼ KR β,In × m ( τ, , I n , I m ) , Re( nβ/ > ( m − β/ − t m . Inaddition, define R = S − / Y where S = S + || Y || . Then S ∼ R β,I (( ν + mn ) β/ P mi =1 t i , k, independent of R . Furthermore the density of R is Γ βm [ nβ/ β [( ν + mn ) β/ k + P mi =1 t i ] π βmn/ Γ βm [ nβ/ , τ ]Γ β [ νβ/ k ] (cid:0) − || R || (cid:1) νβ/ k − q τ ( R ∗ R ) ( d R ) , (10) where (cid:0) − || R || (cid:1) > ; which is termed the standardised matrix multivariate Pear-son type II-Riesz type distribution and is denoted as R ∼ P II R β,Im × n ( ν, k, τ, , , I n , I m ) .Proof. From definition 2.1 and 2.2, the joint density of S and Y is ∝ s βν/ k − etr (cid:8) − β (cid:0) s + || Y || (cid:1)(cid:9) q τ ( Y ∗ Y ) ( ds )( d Y )where the constant of proportionality is c = β νβ/ k Γ β [ νβ/ k ] · β mnβ/ P mi =1 t i Γ βm [ nβ/ π mnβ/ Γ βm [ nβ/ , τ ] . Making the change of variable S = S − || Y || and Y = S / R , by (5)( ds ) ∧ ( d Y ) = s βmn/ ( ds ) ∧ ( d R ) . S = S − || Y || = S (cid:0) − || R || (cid:1) , the joint density of S and R is ∝ (cid:0) − || R || (cid:1) βν/ k − s βν/ k − etr {− βs } q τ ( s R ∗ R ) ( ds )( d R ) . Also, note that q τ ( s R ∗ R ) = q τ (cid:16) ( s / I m ) R ∗ R ( s / I m ) (cid:17) = q τ ( s I m ) q τ ( R ∗ R ) = s P mi =1 t i q τ ( R ∗ R ) . From where, the joint density of S and R is= β ( ν + mn ) β/ k + P mi =1 t i Γ β [( ν + mn ) β/ k + P mi =1 t i ] etr {− βs } s ( ν + mn ) β/ k + P mi =1 t i − ( ds ) × Γ βm [ nβ/ β [( ν + mn ) β/ k + P mi =1 t i ] π βmn/ Γ βm [ nβ/ , τ ]Γ β [ νβ/ k ] (cid:0) − || R || (cid:1) νβ/ k − q τ ( R ∗ R ) ( d R ) , which shows that S ∼ R β,I (( ν + mn ) β/ P mi =1 t i , k, R ,where R has the density (10). Corollary 3.1. Let R ∼ P II R β,Im × n ( ν, k, τ, , , I n , I m ) and define C = ρ − / U ( Θ ) ∗ R U ( Σ ) + µ where U ( B ) ∈ T βU ( n ) , such that B = U ( B ) ∗ U ( B ) is the Cholesky decomposition of B ∈ S βm , Θ ∈ P βn , Σ ∈ P βm , ρ > constant and µ ∈ L βn,m is a matrix of constants.Then the density of S is ∝ (cid:0) − ρ tr Σ − ( C − µ ) ∗ Θ − ( C − µ ) (cid:1) νβ/ k − × q τ (cid:2) U ( Σ ) ∗− ( C − µ ) ∗ Θ − ( C − µ ) U ( Σ ) − (cid:3) ( d S ) (11) where (cid:0) − ρ tr Σ − ( C − µ ) ∗ Θ − ( C − µ ) (cid:1) > ; with constant of proportionality Γ βm [ nβ/ β [( ν + mn ) β/ − k − P mi =1 t i ] ρ mnβ/ − P mi =1 t i π βmn/ Γ βm [ nβ/ , − τ ]Γ β [ νβ/ − k ] | Σ | βn/ | Θ | βm/ , which is termed the matrix multivariate Pearson type II-Riesz distribution and isdenoted as C ∼ P II R β,Im × n ( ν, k, τ, ρ, µ , Θ , Σ ) .Proof. Observe that R = ρ / U ( Θ ) ∗− ( C − µ ) U ( Σ ) − and( d R ) = ρ mnβ/ | Σ | − βn/ | Θ | − βm/ ( d C ) . The desired result is obtained making this change of variable in (10).10ext we derive the corresponding matrix multivariate beta type I distribution . Theorem 3.2. Let R ∼ P II R β,In × m ( ν, k, τ, ρ, , I n , Σ ) , and define B = R ∗ R ∈ P βm ,with n ≥ m . Then the density of B is, ∝ | B | ( n − m +1) β/ − (1 − ρ tr Σ − B ) νβ/ k − q τ ( B )( d B ) , (12) where − ρ tr Σ − B > ; and with constant of proportionality Γ β [( ν + mn ) β/ k + P mi =1 t i ] ρ βmn/ P mi =1 t i Γ βm [ nβ/ , τ ]Γ β [ νβ/ k ] | Σ | nβ/ q τ ( Σ ) . B is said to have a nonstandarised matrix multivariate beta-Riesz type I distribu-tion .Proof. The desired result follows from (10), by applying (7) and then (1); and ob-serving that q τ ( U ( Σ ) ∗− B U ( Σ ) − ) = q − τ ( Σ ) q τ ( B ).In particular if Σ = I m in Theorem 3.2, we obtain: Corollary 3.2. Let R ∼ P II R β,In × m ( ν, k, τ, , , I n , I m ) , and define B = R ∗ R ∈ P βm ,with n ≥ m . Then the density of B is, Γ β [( ν + mn ) β/ k + P mi =1 t i ]Γ βm [ nβ/ , τ ]Γ β [ νβ/ k ] | B | ( n − m +1) β/ − (1 − ρ tr B ) νβ/ k − q τ ( B )( d B ) , (13) where − ρ tr B > . B is said to have a matrix multivariate beta-Riesz type Idistribution . Remark 3.1. Observe that alternatively to classical definitions of generalised ma-tricvariate beta function (for symmetric cones), see D´ıaz-Garc´ıa [5], Faraut andKor´anyi [14] and Hassairi et al. [20], defined as B βm [ a, κ ; b, τ ]= Z < S < I m | B | b − ( m − β/ − q τ ( B ) | I m − B | a − ( m − β/ − q κ ( I m − B )( d B )= Z F ∈ P βm | F | b − ( m − β/ − q τ ( F ) | I m + F | − ( a + b ) q − ( κ + τ ) ( I m + F )( d F )= Γ βm [ a, κ ]Γ βm [ b, τ ]Γ βm [ a + b, κ + τ ] , where κ = ( k , k , . . . , k m ) ∈ ℜ m , τ = ( t , t , . . . , t m ) ∈ ℜ m , Re( a ) > ( m − β/ − k m and Re( b ) > ( m − β/ − t m . From Corollary 3.2 and D´ıaz-Garc´ıa and Guti´errez-S´anchez [10, Theorem 3.3.1], we have the following alternative definition:11 efinition 3.1. The matrix multivariate beta function is defined an denoted as: B ∗ βm [ a, k ; b, τ ] = Z − tr B > | B | b − ( m − β/ − (1 − tr B ) a + k − q τ ( B )( d B )= Z R ∈ P βm | F | b − ( m − β/ − (1 + tr F ) − ( a + mb + k + P mi =1 t i ) q τ ( F )( d F )= Γ β [ a + k ]Γ βm [ b, τ ]Γ β [ a + mb + k + P mi =1 t i ] . Also, observe that, when m = 1, then τ = t and κ = k and B β [ a, k ; b, t ] = Γ β [ a + k ]Γ β [ b + t ]Γ β [ a + b + k + t ] = B ∗ β [ a, k ; b, t ]Finally observe that if in results in this section are defined k = 0 and τ =(0 , . . . , In this section, the joint densities of the singular values of random matrix R ∼P II R β,In × m ( ν, k, τ, , , I n , I m ) are derived. In addition, and as a direct consequence,the joint density of the eigenvalues of matrix multivariate beta-Riesz type I distri-bution is obtained for real normed division algebras. Theorem 4.1. Let δ , . . . , δ m , > δ > · · · > δ m > , be the singular values of therandom matrix R ∼ P II R β,In × m ( ν, k, τ, , , I n , I m ) . Then its joint density is m π βm / ̺ Γ βm [ βm/ B ∗ βm [ νβ/ , k ; nβ/ , τ ] m Y i =1 (cid:0) δ i (cid:1) ( n − m +1) β/ − / − ρ m X i =1 δ i ! νβ/ k − × m Y i 0, where G = diag( λ , . . . , λ m ). As visual examples, different Pearson type II-Riesz densities for m = 1 are showedin figures 1 and 2,Recall that in octonionic case, from the practical point of view, we most keep inmind the fact from Baez [1], there is still no proof that the octonions are useful forunderstanding the real world . We can only hope that eventually this question will besettled on one way or another. In addition, as is established in Faraut and Kor´anyi[14] and Sawyer [28] the result obtained in this article are valid for the algebra ofAlbert , that is when hermitian matrices ( S ) or hermitian product of matrices ( X ∗ X )are 3 × Acknowledgements This paper was written during J. A. D´ıaz-Garc´ıa’s stay as a visiting professor atthe Department of Statistics and O. R. of the University of Granada, Spain; it staywas partially supported by IDI-Spain, Grants No. MTM2011-28962 and under theexisting research agreement between the first author and the Universidad Aut´onomaAgraria Antonio Narro, Saltillo, M´exico.13 .0 0.2 0.4 0.6 0.8 1.0 Pearson II−Riesz type density functions x f ( x ) k=6k=4k=2 Figure 1: With ν = 15, n = 18 and t = 7 Pearson II−Riesz type density functions x f ( x ) t=7t=5t=3 Figure 2: With ν = 3, n = 18 and k = 014 eferences [1] Baez, J. C. (2002). The octonions, Bull. Amer. Math. Soc. 39, 145–205.[2] Boutouria. I., and Hassiri. A. (2009). 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