Page 44 - Textos de Matemática Vol. 34

34 AM´ILCAR BRANQUINHO
We consider an N ×N positive definite matrix of measures, W (i.e., for any Borel set A ⊂ R , W (A) is a positive semidefinite numerical matrix), having moments of

every order (i.e., the matrix integral tn d W (t) exists for all n ∈ N ). R
Assuming that P (t) d W (t)P ∗(t) , where P ∗ means the conjugate transpose
operation over the matrix P , is nonsingular for any matrix polynomial P with non- singular leading coefficient, it can be shown that the matrix inner product defined by W in the linear space of matrix polynomials, has a sequence of orthonormal matrix polynomials {Pn} defined by

Pn(t)dW(t)Pm∗ (t) = δn,mI , n,m ∈ N,
Note that, {Pn} defined in this way is, for each n ∈ N a matrix polynomial of
degree n, i.e.
Pn(t)=P0,ntn +P1,ntn−1 +···+Pn,n, n∈N,
and is unique up to a multiplication on the left by a unitary matrix, where Pj,k are forj=0,1,...,n,N×N matricesandP0,n isnonsingularforalln∈N.
There is a analogous of the Favard’s theorem for the sequence of orthonormal matrix polynomials, {Pn} (see for instance [1, 5]), and so {Pn} verifies a three term recurrence relation
(1.1) tPn(t) = An+1Pn+1(t) + BnPn(t) + A∗nPn−1(t) , n ∈ N ,
where P−1 is the null (N × N)-matrix, θ, and P0 ∈ CN×N \ {θ} , An are nonsin- gular matrices and Bn are hermitean matrices. Because the polynomials Qn(t) = UnPn(t) , with UnUn∗ = I , for n ∈ N are also orthonormal with respect to the same measure W, it can be proved that they satisfy a three term recurrence relation as (1.1), where instead of the matrix coefficients An and Bn we have Un−1AnUn∗ and UnBnUn∗ , respectively.
In section 2 we use some properties of the zeros of matrix polynomials {Pn} , i.e. the zeros of det Pn . This means that Pn has nN zeros taking into account their multiplcities.
The three term recurrence relation and Christoffel-Darboux identity for matrix orthogonality are shown to be a useful tool for solving certain problems in the scalar orthogonality (cf. [4, section 5]), in network models (cf. [2]), and is an added value to the theory of scalar orthogonal polynomials (see [6]).
2. Christoffel-Darboux identity
Theorem 2.1. Let {Pn} be a sequence of matrix polynomials, such that for each n, deg Pn = n . Then the following condition are equivalent: