Jordan structures of irreducible totally nonnegative matrices with a prescribed sequence of the first p-indices

Let A∈Rn×n\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$A \in {\mathbb {R}}^{n \times n}$$\end{document} be an irreducible totally nonnegative matrix with rank r and principal rank p, that is, A is irreducible with all minors nonnegative, r is the size of the largest invertible square submatrix of A and p is the size of its largest invertible principal submatrix. We consider the sequence {1,i2,…,ip}\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\{1,i_2,\ldots ,i_p\}$$\end{document} of the first p-indices of A as the first initial row and column indices of a p×p\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$p \times p$$\end{document} invertible principal submatrix of A. A triple (n, r, p) is called (1,i2,…,ip)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(1,i_2,\ldots ,i_p)$$\end{document}-realizable if there exists an irreducible totally nonnegative matrix A∈Rn×n\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$A \in {\mathbb {R}}^{n \times n}$$\end{document} with rank r, principal rank p, and {1,i2,…,ip}\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\{1,i_2,\ldots ,i_p\}$$\end{document} is the sequence of its first p-indices. In this work we study the Jordan structures corresponding to the zero eigenvalue of irreducible totally nonnegative matrices associated with a triple (n, r, p) (1,i2,…,ip)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(1,i_2,\ldots ,i_p)$$\end{document}-realizable.


Introduction
A matrix A ∈ R n×n is called totally nonnegative (totally positive) if all its minors are nonnegative (positive) and it is abbreviated as TN (TP). These classes of matrices have been studied by several authors [1,[8][9][10]13] obtaining properties, the Jordan structure and characterizations by applying the Gaussian or Neville elimination with applications in algebra, geometry, differential equations, economics and other fields. We recall that A ∈ R n×n , n ≥ 2, is an irreducible matrix if there is not a permutation matrix P such that P AP T where O is an (n − q) × q zero matrix (1 ≤ q ≤ n − 1)). If n = 1, In [8, p. 87] the authors denoted by ITN the irreducible TN matrices. The rank of A, denoted by rank(A), is the size of the largest invertible square submatrix of A. The principal rank of A, denoted by p-rank(A), is the size of the largest invertible principal submatrix of A. If there exists an ITN matrix A ∈ R n×n with rank(A) = r and p-rank(A) = p, then the triple (n, r , p) is called realizable [10, p. 709], and A is considered as an ITN matrix associated with the triple (n, r , p). Since the nonzero eigenvalues of A are positive and distinct [10,Theorem 3.3], the set of eigenvalues {λ i } n i=1 of A satisfies that λ 1 > λ 2 > · · · > λ p > 0, and λ p+1 = λ p+2 = · · · = λ n = 0, that is, if A is an associated matrix with a realizable triple (n, r , p) then p is the number of nonzero eigenvalues of A and n − p is the algebraic multiplicity of its zero eigenvalue. The matrix A has n −r zero Jordan blocks whose sizes are given by the Segre characteristic of A corresponding to its zero eigenvalue [14]. Moreover, since rank(A p ) = p the size of its zero Jordan blocks is at most p. Now, we consider the next definition.
Definition 1 A Jordan structure corresponding to the zero eigenvalue (zero-Jordan structure) is called admissible for a realizable triple (n, r , p) if there exists an ITN matrix A ∈ R n×n with rank(A) = r , p-rank(A) = p, and A has the given Jordan structure.
From this definition, if A ∈ R n×n is an ITN matrix associated with the realizable triple (n, r , p) and S = (s 1 , s 2 , . . . , s n−r ) is the Segre characteristic corresponding to its zero eigenvalue, then this Segre characteristic is a zero-Jordan structure admissible for (n, r , p). That is, the zero-Jordan structure admissible for the realizable triple (n, r , p) is the same concept as the Segre characteristic corresponding to the zero eigenvalue of an ITN matrix associated with the realizable triple (n, r , p) . We will talk about zero-Jordan blocks if we do not need to specify the size of the Jordan blocks, and about the Segre characteristic in the opposite case. Now, our question is how to be sure if a decreasing sequence of positive integers gives a zero-Jordan structure admissible for a realizable triple. The answer is that the sequence denoted by S = (s 1 , s 2 , . . . , s q ) is a zero-Jordan structure admissible for a realizable triple (n, r , p) if s i represents the size of each zero Jordan block, i = 1, 2, . . . , q and the following inequalities hold: . . , n − r n−r i=1 s i = n − p. The problem to characterize completely all possible Jordan structures admissible for a realizable triple has been studied by several authors (see for instance [5,8,10]) extending the classical result of [11] who introduced the total positivity and studied the eigenvalues of the oscillatory matrices. These results are related to the relationship between the Jordan structure of two matrices sufficiently close, deflation problems, the pole assignment problem and the stability in control systems (see [2] and references therein).
In [5] the authors have obtained the number of zero-Jordan structures admissible for a realizable triple (n, r , p), the Segre characteristics corresponding to these zero-Jordan structures and an algorithm to compute them. Since the zero-Jordan structures admissible for a realizable triple (n, r , p) can be interpreted as the number of partitions of n − p into exactly n − r parts with the largest part at most p, then the following properties hold: 1. If (n, r , p) is a realizable triple and S = (s 1 , s 2 , . . . , s n−r −1 , s n−r ) is a zero-Jordan structure admissible for this triple, then S = (s 1 , s 2 , . . . , s n−r −1 , s n−r , 1) is a zero-Jordan structure admissible for the realizable triple (n + 1, r , p). 2. If (n, r , p) is a realizable triple and S = (s 1 , s 2 , . . . , s n−r −1 , s n−r ) is a zero-Jordan structure admissible for this triple, we have the following options: (a) If s n−r = 1, then S = (s 1 , s 2 , . . . , s n−r −1 ) is a zero-Jordan structure admissible for the realizable triple (n − 1, r , p).
We follow the notation of [1], that is, for k, n ∈ N, 1 ≤ k ≤ n, Q k,n denotes the set of all increasing sequences of k natural numbers less than or equal to n.
Note that if A is TN without null rows or columns, then i 1 = 1. In [4,Section 2] the authors use this sequence to study the linear dependence relations between rows or columns of TN matrices. . . , i p } as the sequence of its first p-indices.

Definition 3 [4, Definition 2]
If a matrix A satisfies the conditions of Definition 3 we say that A is a matrix associated with the triple (n, r , p) (1, i 2 , . . . , i p )-realizable.
We recall that a matrix is an upper block echelon matrix if the first nonzero entry in each row (leading entry) is to the right of the leading entry in the row above it and all zero rows are at the bottom. A matrix is upper block echelon if each nonzero block, starting from the left, is to the right of the nonzero blocks below and the zero blocks are at the bottom. A matrix is a lower (block) echelon matrix if its transpose is an upper (block) echelon matrix.
In [6] is presented an algorithm to obtain an upper block echelon TN matrix U of size n × n, with rank(U ) = r , p-rank(U ) = p and one of the zero-Jordan structure admissible for the triple (n, r , p). This algorithm also computes the sequence of the first p-indices of U . With the obtained matrix U the authors construct an ITN matrix A associated with the realizable triple (n, r , p) and with the same zero-Jordan structure and sequence of the first p-indices as matrix U .
On the other hand, if we add the sequence of the first p-indices H = {1, i 2 , i 3 , . . . , i p } to the triple (n, r , p) in [7] is given an algorithm to obtain an upper block echelon TN matrix U of size n × n, with rank(U ) = r , p-rank(U ) = p and with H as its sequence of the first p-indices. From this matrix U the authors construct an ITN matrix A associated with the triple (n, r , p)(1, i 2 , . . . , i p )-realizable.
The difference between prescribing or not the sequence of the first p-indices is that some properties that the ITN matrices satisfy without prescribed p-indices, are not satisfied when they are prescribed. For instance: 1. By [9], the upper bound for the maximum rank of an ITN matrix A associated with a realizable triple (n, r , p) is n − n − p p , but this bound can be lower when the sequence of the first p-indices is prescribed (see [4]  In this work we obtain the zero-Jordan structures admissible for a realizable triple (n, r , p) with a sequence of the first p-indices {1, i 2 , . . . , i p } and we give a method to construct an ITN matrix associated with this triple (n, r , p) (1, i 2 , . . . , i p )-realizable and with one of the zero-Jordan structure admissible. For that, in Sect. 2 we study some properties of the zero-Jordan structures for an upper block echelon TN matrix U ∈ R n×n , with rank(U ) = r , p-rank(U ) = p in function of its sequence of the first p-indices. In Sect. 3 we construct Algorithm 4 to obtain how many and what are explicitly the zero-Jordan structures admissible for a triple (n, r , p)(1, i 2 , . . . , i p )-realizable. In Sect. 4 we present a procedure to obtain paths of pairs associated with the zero-Jordan structure obtained in Sect. 3 and, using these paths we present an algorithm to obtain an upper block echelon TN matrix U ∈ R n×n , with rank(U ) = r , p-rank(U ) = p, a prescribed sequence of its first p-indices and with one of the zero-Jordan structures. Finally, we obtain some matrices associated with a triple (n, r , p) (1, i 2 , . . . , i p )-realizable and with one of the zero-Jordan structures obtained in Sect. 3.
From now on and for simplicity, we use the following MatLab notation: A(i, :) denotes the ith row of A and A(:, j) denotes its jth column; ones(n, m) denotes the n × m matrix of ones; triu(ones(n, m)) denotes the upper triangular part of ones(n, m); zeros(n, m) denotes the n × m zero matrix; diag(v) denotes a square matrix of order n, with the elements of v on the main diagonal, where v is a vector of n components; tril(ones)(n, n) denotes the lower triangular part of ones(n, n).

Properties on the zero-Jordan structure of the upper block echelon TN matrices
In this section we study some properties of the zero-Jordan structures for an upper block echelon TN matrix in function of the sequence of its first p-indices. Concretely, we consider the following upper block echelon TN matrix U ∈ R n×n , with rank(U ) = r , p-rank(U ) = p and H = {1, 2, . . . , j + 1, i j+2 , . . . , i p }, i j+2 > j + 2, as the sequence of its first p-indices (1) whose partition in blocks by rows and columns is, respectively 0 0 · · · 1 1 · · · 1 1 ⎤ ⎥ ⎥ ⎥ ⎦ = triu(ones( j + 1, n)).
Following the steps (a), (b) and (c) of the process described in [6, Theorem 1], we transform the matrix U , by similarity and permutation similarity, into the matrix T = XU X −1 where the block partition by rows and columns is is obtained as a product of matrices, being one of them the permutation matrix . . , e j+1 , e i j+2 , . . . , e i p , e j+2 , . . . , e i j+2 −1 , e i j+2 +1 , . . . , e n where e i is the ith vector of the canonical basis of R n .

Remark 1
Consider the matrix T given in (2), 1. T 11 ∈ R p× p is a non-derogatory nonsingular matrix, with one Jordan block of size p corresponding to its unique eigenvalue λ = 1. 2. T 2 is a nilpotent matrix. The number of blocks T s,s+1 of the superdiagonal of T 2 depends on the number and the distribution of the p-indices that U has after the j first consecutive indices.
As a result, the maximum size of the zero Jordan blocks of T 2 , and therefore of U , is p − j. . . , i p }, then T 2 has r + w blocks in the superdiagonal, being w the number of nonconsecutive indices.
In this case, the maximum size of the zero Jordan blocks of U is r + w + 1. 3. Since U and T are similar we have that the zero-Jordan structure of U and T 2 is the same. Therefore, for simplicity, we use T 2 to study the zero-Jordan structure of U .

Example 1 Consider the upper block echelon TN matrix
By similarity we obtain the matrix From Remark 1, T 11 ∈ R 5×5 is nonsingular and has a Jordan block of size p = 5 corresponding to its unique eigenvalue λ = 1. Moreover, since there are nonconsecutive indices in {7, 9}, T 2 has p − ( j + 1) = 5 − (2 + 1) = 2 blocks in the superdiagonal, and the sizes of T 23 and T 34 are (i 4 respectively. So, the maximum size of the zero Jordan blocks of U is p − j = 5 − 2 = 3. In this case, from T 2 we conclude that the Segre characteristic of U corresponding to its zero eigenvalue is S = (2, 2, 2).
Note that, if we apply [5, Algorithm 3] to the realizable triple (11,8,5) we have that the zero-Jordan structures admissible are S 1 = (4, 1, 1), S 2 = (3, 2, 1) and S 3 = (2, 2, 2). Next, we use [6, Algorithm 3] for each zero-Jordan structure admissible and we construct an upper block echelon TN matrix U , of size 11 × 11, rank(U ) = 8, p-rank(U ) = 5 and with one of these 3 zero-Jordan structures. Now, if we add H = {1, 2, 3, 7, 9} as the sequence of the first 5-indices, we have shown that the maximum size of the zero Jordan blocks of U is 3. Then, we can conclude that does not exist an upper block echelon TN matrix U of size 11 × 11, rank(U ) = 8, p-rank(U ) = 5, H = {1, 2, 3, 7, 9} as the sequence of its first 5-indices and with the Segre characteristic corresponding to its zero eigenvalue given by S 1 . . . , i p } as the sequence of its first p-indices, with i j+2 > j + 2, and S = (s 1 , s 2 , . . . , s n−r ) as the Segre characteristic corresponding to its zero eigenvalue. Then, there exists an upper block echelon TN . . , i p − j} as the sequence of the first ( p − j)-indices, whose Segre characteristic corresponding to the zero eigenvalue is S.

Theorem 1 Let U ∈ R n×n be an upper block echelon TN matrix with rank
Proof Let U ∈ R n×n be an upper block echelon TN matrix given in (1) with rank(U ) = r , p-rank(U ) = p and H = {1, 2, . . . , j +1, i j+2 , . . . , i p } as the sequence of its first p-indices, i j+2 > j + 2.
From the matrix U , we consider the TN matrixŨ ∈ R (n− j)×(n− j) , such that, rank and whose partition by blocks in rows and columns is, respectively

Now, we consider the similarity matrix
where X is the matrix given in (3) andX 12 = X 12 ( j + 1, :). Taking into account the structure of the matrices X andX , we obtaiñ being the partition by blocks in rows and columns From similarity, the zero-Jordan structure ofŨ is equal to the zero-Jordan structure of the nilpotent matrix T 2 , which is equal to the zero-Jordan structure of the matrix U . Thus, the Segre characteristic ofŨ ∈ R (n− j)×(n− j) corresponding to its zero eigenvalue is equal to  . . , i p − j} as the sequence of the first ( p − j)-indices, whose Segre characteristic corresponding to the zero eigenvalue is S.
Proof From the matrix U and applying similarity we obtain the matrix T with the partition by blocks of rows and columns Note that, the submatrix T 2 can be obtained from an upper block echelon TN matrix . . , i p − j} as the sequence of its first ( p − j)-indices. Then, the Segre characteristic corresponding to the zero eigenvalue of U andŨ is the same. 1. Applying Theorem 1 we may assume, without loss of generality, that the first elements of the sequence of the first p-indices of the matrix U are nonconsecutive. That is, i 2 > 2. 2. Applying Theorem 2 to each group of consecutive indices we can suppose, without loss of generality, that in the sequence of the first p-indices of the matrix U there are not groups of consecutive entries.    . . , s n−r ) as the Segre characteristic corresponding to its zero eigenvalue. Then, there exists an upper block echelon TN matrixŨ ∈ R (n+ j)×(n+ j) with rank(Ũ ) = r + j, p-rank(Ũ ) = p + j and H = {1, 2, . . . , j, 1 + j, i 2 + j, i 3 + j, . . . , i p + j} as the sequence of the first ( p − j)-indices, whose Segre characteristic corresponding to the zero eigenvalue is S.
From the matrix U , we consider the TN matrixŨ ∈ R (n+ j)×(n+ j) , such that, rank By construction it is satisfied that Then, for t = 1, 2, 3, . . . , p, So, the zero-Jordan structure ofŨ is equal to the zero-Jordan structure of U . Thus, the Segre characteristic ofŨ corresponding to its zero eigenvalue is equal to S = (s 1 , s 2 , . . . , s n−r ).
The following two results show how the rank, the principal rank, the sequence of the first p-indices and the Segre characteristic of an upper block echelon TN matrix change when its size increases by one unit.  . . , i p } as the sequence of its first p-indices and whose Segre characteristic corresponding to its zero eigenvalue is S = (s 1 , s 2 , . . . , s n−r , 1).
Proof Consider the following upper block echelon TN matrix U ∈ R n×n , with rank(U ) = r , p-rank(U ) = p, H = {1, i 2 , i 3 , . . . , i p } as the sequence of its first p-indices .
where the block partition by rows and columns is, respectively

By similarity we obtain
whose block partition by rows and columns is As the Jordan structure of T 2 is the same as the zero-Jordan structure of U , the Segre characteristic corresponding to its zero eigenvalue is S = (s 1 , s 2 , . . . , s n−r ). Now, we consider the upper block echelon TN matrixŨ ∈ R (n+1)×(n+1) , with rank(Ũ ) = r , p-rank(Ũ ) = p, H = {1, i 2 , i 3 , . . . , i p } as the sequence of its first p-indices given bỹ .
Applying the similarityXŨX −1 =T , wherẽ we obtain the matrix Applying the similarity Thus, the Segre characteristic ofŨ corresponding to its zero eigenvalue is S = (s 1 , s 2 , . . ., s n−r , 1).  . . , i p , n + 1} as the sequence of its first ( p + 1)-indices and whose Segre characteristic corresponding to its zero eigenvalue is S.

Remark 3 Applying
Proof Consider the matrix U and construct the following upper block echelon TN matrix, . . , i p , n + 1} as the sequence of its first ( p + 1)-indices, Applying the similarity XU X −1 we obtain the matrix T given in (4), being the block partition by rows and columns whose block partition by rows and columns is From similarity, the Jordan structure of T 2 is the same as the zero-Jordan structure of U . Thus, the Segre characteristic corresponding to the zero eigenvalue of U andŨ is the same.

. , i p )-realizable
In [5] given a realizable triple (n, r , p) the authors obtain the number of the zero-Jordan structures admissible for a realizable triple (n, r , p), these zero-Jordan structures and an algorithm to compute them. Now, as noted in Sect. 1 a key objective is to obtain the zero-Jordan structures admissible for a triple (n, r , p) (1, i 2 , . . . , i p )-realizable. That is, we want to answer how many and what are the zero-Jordan structures admissible for a triple (n, r , p) (1, i 2 For that, we consider an upper block echelon TN matrix U ∈ R n×n with rank(U ) = r , p-rank(U ) = p and H = {1, i 2 , i 3 , . . . , i p } as the sequence of its first p-indices. Recall that in Sect. 2 we transform by similarity the matrix U into the matrix T given in (2). This matrix T has a nilpotent upper block echelon matrix T 2 of size (n − p)×(n − p) with rank r − p, and whose block partition by rows and columns is {i 2 − 2, i 3 − i 2 − 1, . . . , i p − i p−1 − 1, n − i p } and whose zero-Jordan structure is the same as the zero-Jordan structure of U (see Remark 1).
Given that the zero-Jordan structures admissible for a triple (n, r , p) (1, i 2 , i 3 , . . . , i p )realizable are the same as the zero-Jordan structures of an upper block echelon TN matrix U of size n × n, with rank(U ) = r , p-rank(U ) = p and (1, i 2 , . . . , i p ) as the sequence of the first p-indices, we are going to study the different zero-Jordan structures that the matrix T 2 admits. These structures are obtained by Algorithm 4. This algorithm needs to know all possible linearly independent combinations of r − p rows of T 2 (note that the rows are between the first one and the row i p − p). For each linearly independent combination of r − p rows we apply Algorithm 3, which computes the zero-Jordan structures taking into account all possible linearly independent combinations of r − p columns (note that the columns are between the column i 2 − 1 and the last column). Algorithm 3 needs two auxiliary algorithms to run correctly. Thus, we present these two algorithms.
Algorithm 1 obtains the conjugated sequence of a given sequence of nonincreasing positive integers.

Obtaining upper block echelon TN matrices
In this section we present a method to construct an upper block echelon TN matrix U ∈ R n×n with rank(U ) = r , p-rank(U ) = p, H = {1, i 2 , . . . , i p } as the sequence of its first p-indices and S = (s 1 , s 2 , . . . , s n−r ) as the Segre characteristic corresponding to its zero eigenvalue. With the following process we obtain paths of pairs, P z = {(i, j)}, z = 1, 2, . . ., associated with the zero-Jordan structure obtained in Algorithm 4. These paths are used to construct the upper block echelon TN matrix U . (n, r , p) . . associated with these zero-Jordan structures. 8. 1. for all P z , z = 1, 2, . .

. obtained in previous Steps, match the inputs of I t with other
inputs of its corresponding J t , taking into account that the pairs whose j ∈ H cannot be modified and each j must be greater than its equivalent j in P z ; 8.2. obtain P z = {(i, j)}, z = 3, 4, . . . and go Step 10. 9. Use the diagram for P 2 obtained in Step 7, 9.1. denoted by a m , m = 1, 2, . . . the vertices that are at the bottom of the diagram and choose a m ≤ i p ; 9.2. denoted by b m , m < b m , m = 1, 2, . . ., the vertices that do not appear in the diagram; 9.3. create the pairs (a m , b m ), a m < b m that preserve the ascending order of the path; 9.4. choose the pairs in the diagram such that j / ∈ H and replace each pair by (a m , b m ), m = 1, 2, . . .; 9.5. obtain P z = {(i, j)}, z = 3, 4, . . . and go Step 10. 10. Create the sets I P z and J P z , for each path P z = {(i, j)}, z = 1, 2, . .
We only calculate S from the paths P z obtained previously.
Note that if we use Theorem 1, we reduce the problem to the triple (7, 5, 3) (1, 3, 5)realizable and applying Theorem 3 we obtain the same result.
Remark 5 By using the matrix U calculated in this section, we obtain some matrices associated with this triple and with one of the zero-Jordan structure obtained in Sect. 3. For that, we consider the ITN matrices presented in Sect. 1 and the totally nonpositive matrices, denoted as t.n.p., given in [7].
Recall that a matrix A ∈ R n×n is called type-I t.n.p. matrix if all its minors are nonpositive and −a 11 < 0 and it is called type-II t.n.p. matrix if all its minors are nonpositive, a 11 = 0, −a 12 < 0 and −a 21 < 0.
In the same way as in ITN matrices, a triple (n, r , p) is called (1, i 2 , . . . , i p )-negatively realizable of the type-I (type-II) if there exists a type-I (type-II) t.n.p. matrix A = (−a i j ) ∈ R n×n with rank(A) = r , p-rank(A) = p, and {1, i 2 , . . . , i p } (i 2 = 2) as the sequence of its first p-indices.
We present Algorithm 6 to construct some matrices associated with the realizable triple  (n, r , p). Concretely, we calculate an ITN matrix A 1 , a type-I t.n.p. matrix A 2 and a type-II t.n.p. matrix A 3 using the same matrix U obtained in Algorithm 5. All these matrices have the same sequence of its first p-indices and the same zero-Jordan structure (see [6,7]). Note that in Algorithm 6 we give a value to t, but in [7,Proposition 8] we can see that t is a number such that satisfies t ≥ i p j=2 u jn .
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