Ando, T., Totally positive matrices, Lin. Alg. and Appl. 90 (1987), 165--219.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....= x i (a; t 1 ; t m ) for some sequence i, with all the t k positive and a 2 H 0 . For the type A r (i.e. for G = SLn (C ) n = r 1) a theorem of C. Loewner [16] based on a result by A. Whitney [23] tells that the above definition of total nonnegativity coincides with the usual one [2, 15]: a matrix (with determinant 1) is totally nonnegative if and only if all its minors are nonnegative. The set G0 is the disjoint union of the subsets G 0 obtained by intersecting it with double Bruhat cells: We call the G 0 totally positive varieties; they will be one of the main objects ....

T. Ando, Totally positive matrices, Linear Algebra Appl. 90 (1987), 165-219.

....to be real. We present evidence (computational and Theorems) in support of it. Subsequent sections describe the conjecture in greater generality for enumerative problems arising from the Schubert calculus on Grassmannians in Section 3 and a newer extension involving totally positive matrices [Ando 1987] in Section 4. We describe and give evidence for each extension and show how the version of the conjecture in Section 2 implies more general versions involving Pieri type enumerative problems. In Section 5, we present a counterexample to their original conjecture and discuss further questions. A ....

T. Ando, "Totally positive matrices", Linear Algebra Appl. 90 (1987), 165--219.

....(resp. positive) These matrices appear in various branches of mathematics and its applications, e.g. in mechanics [8] statistics [15] combinatorics [7] and computer aided geometric design [4] to name only a few. For a thorough presentation of the properties of these matrices up to 1984 see [1]. Some more recent results can be found in [11] Some present research focusses on completion problems, cf. 6] A class of real matrices which is related to this class is the inverse nonnegative matrices; these are nonsingular matrices whose inverses are entrywise nonneg Email address: ....

T. Ando, Totally positive matrices, Linear Algebra Appl. 90 (1987) 165--219.

....elements C (M) i along each (sub)diagonal are equal: C (M) i = C (M) j, j i) C (M) j i, j) B.2) we have denoted here the rows and columns of C (M) by a pair of indices in parentheses to distinguish them from the diagonal index in Eq. 4) The following theorem holds [76,78] for an infinite Toeplitz matrix T (#) T (#) is totally positive if and only if its generating function: f(z) # # n= # T (#) n z n (B.3) is of the form f(z) Cz r exp # # 1 z # 1 z # # # 1 (1 # n z) # # 1 (1 # n z) # # 1 (1 # n z) # # 1 (1 # ....

T. Ando, Totally positive matrices, Lin. Alg. Appl. 90 (1987) 165--219.

....frequency sequences [28, 40] representation theory of the infinite symmetric group and the Edrei Thoma theorem [13, 44] planar resistor networks [11] unimodality and log concavity [42] and theory of immanants [43] Further references can be found in S. Karlin s book [28] and in the surveys [2, 5, 38]. In this paper, we focus on the following two problems: i) parametrizing all totally nonnegative matrices; ii) testing a matrix for total positivity. Our interest in these problems stemmed from a surprising representation theoretic connection between total positivity and canonical bases for ....

T. Ando, Totally positive matrices, Linear Algebra Appl. 90 (1987), 165--219.

....; i p j 1 ; j q the submatrix of A consisting of rows i 1 ; i p and columns j 1 ; j q . Given vectors v 1 ; v k 2 IR d we let V be the d Theta k matrix whose j th column is v j treated as a column vector (v 1 j ; v d j ) T . Following Ando [ 1 ] let Q k;d denote the set of all k tuples ff = ff 1 ; ff k ) such that 1 ff 1 ff 2 Delta Delta Delta ff k d. It is shown in [ 11 ] pages 2 280 281, that the dimension of the space Omega k (IR d ) is Gamma d k Delta and a basis for the space is given by the ....

....0 B 1 p ff 1 (s 0 ) p ff k (s 0 ) 1 p ff 1 (s k ) p ff k (s k ) 1 C A = M u 0 ; un s 0 ; s k 0 B 1 P ff 1 0 : P ff k 0 . 1 P ff 1 n : P ff k n 1 C A By applying the Cauchy Binet formula (see [ 1 ], formula (1.23) and making the substitution (2.2) we obtain (OE ff ) 0 (p(s 0 ) p(s k ) X 0fi0 fi1 Delta Delta Delta fi kn det M u fi 0 ; u fi k s 0 ; s k (OE ff ) 0 (P fi 0 ; P fi k ) 2:4) For general OE 2 Omega k (IR d ) there exist ....

Ando, T. Totally positive matrices, Linear Algebra Appl. 90 (1987), 165--219.

....of matrices that will be used in this paper. An n n matrix A is TP k if all r r minors of A are nonnegative for all r = 1, k. If A is TP n , then it is called totally positive. This class of matrices has many applications in mathematics, statistics, economics, etc. see [14] [1]) Some recent characterizations of totally positive matrices can be found in [9] 10] 11] In Proposition 2.1 we prove that a symmetric TP 2 matrix A with nonzero rows is p banded if and only if c(G(A) # p; so, in particular, G(A) is triangle free if and only if A is tridiagonal. The ....

T. ANDO, Totally positive matrices, Linear Algebra Appl., 90 (1987), pp. 165--219.

....to be real. We present evidence (computational and Theorems) in support of it. Subsequent sections describe the conjecture in greater generality for enumerative problems arising from the Schubert calculus on Grassmannians in Section 3 and a newer extension involving totally positive matrices [1] in Section 4. We describe and give evidence for each extension and show how the version of the conjecture in Section 2 implies more general versions involving Pieri type enumerative problems. In Section 5, we present a counterexample to their original conjecture and discuss further questions. A ....

T. Ando, Totally positive matrices, Lin. Alg. Appl., 90 (1987), pp. 165--219.

....= x i (a; t 1 ; t m ) for some sequence i, with all the t k positive and a 2 H 0 . For the type A r (i.e. for G = SLn (C ) n = r 1) a theorem of C. Loewner [16] based on a result by A. Whitney [23] tells that the above definition of total nonnegativity coincides with the usual one [2, 15]: a matrix (with determinant 1) is totally nonnegative if and only if all its minors are nonnegative. The set G0 is the disjoint union of the subsets G u;v 0 obtained by intersecting it with double Bruhat cells: G u;v 0 = G0 G u;v : We call the G u;v 0 totally positive varieties; they ....

T. Ando, Totally positive matrices, Linear Algebra Appl. 90 (1987), 165-219.

....[30] in 1930, in a paper where he considers some variation diminishing properties of such matrices, and their generalizations. Spectral Properties 29 The main theorem, which we now state, is to be found in Gantmacher, Krein [11] Theorems 10 and 14, p. 460 467) A proof also appears in Ando [1], Gantmacher [8] and Gantmacher, Krein [13] Theorem 5.2. The n eigenvalues of an n Theta n oscillation matrix are positive and simple. In addition, if we denote by u k a real eigenvector (unique up to multiplication by a non zero constant) associated with the eigenvalue k , where 1 2 ....

Ando, T., Totally positive matrices, Lin. Alg. and Appl. 90 (1987), 165-- 219.

....equations and briefly mentioned in [Sh] Later talking to F. Brenti the authors were surprised to find out that this result and its corollary have apparently escaped the attention of the specialists although very closely related theorems proved by more or less the same methods are quoted in [An] and especially in [Cr] ....

T. Ando, Totally positive matrices, Linear Algebra Appl 90 (1987), 165--220.

....1 for j = 1; n. In addition, the eigenvalues of the principal submatrices obtained by deleting the first (or last) row and column of A strictly interlace those of the original matrix. Proofs of this result may be found in Gantmacher, Krein [4] in Gantmacher, Krein [5] and in Ando [1]. A very important property of STP matrices is that of variation diminishing. This was Schoenberg s initial contribution to the theory. The particular result we need may be found in Gantmacher, Krein [5] and in Karlin [6] and is the following. Theorem B. Let A be an n Theta m STP matrix. Then ....

Ando, T., Totally positive matrices, Lin. Alg. and Appl. 90 (1987), 165--219.

....determinant in (7) along the first row, we see that we always obtain two equal columns. With Laplace expansion into 2 by 2 minors we obtain in bracket notation a straightening syzygy: det 2 6 6 4 x 11 x 12 0 0 x 21 x 22 x 21 x 22 x 31 x 32 x 31 x 32 x 41 x 42 x 41 x 42 3 7 7 5 j 0: 1 2][3 4] Gamma [1 3] 2 4] 1 4] 2 3] 0: 7) We can rewrite the nonstandard monomial [1 4] 2 3] as a linear combination of standard ones. The defining ideal of the Grassmann manifold is generated by all straightening syzygies needed to rewrite all possible nonstandard tableaux. In [37] it is proven ....

....in (7) along the first row, we see that we always obtain two equal columns. With Laplace expansion into 2 by 2 minors we obtain in bracket notation a straightening syzygy: det 2 6 6 4 x 11 x 12 0 0 x 21 x 22 x 21 x 22 x 31 x 32 x 31 x 32 x 41 x 42 x 41 x 42 3 7 7 5 j 0: 1 2] 3 4] Gamma [1 3][2 4] 1 4] 2 3] 0: 7) We can rewrite the nonstandard monomial [1 4] 2 3] as a linear combination of standard ones. The defining ideal of the Grassmann manifold is generated by all straightening syzygies needed to rewrite all possible nonstandard tableaux. In [37] it is proven that these ....

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T. Ando. Totally positive matrices. Linear Algebra. Appl., 90:165--219, 1987.

....parameterization that permits minors to be evaluated to high relative accuracy. We give many examples of this below. The rest of this section is organized as follows. First, we give several examples of TP matrices and their parameterizations where high accuracy formulas for their minors exist [2, 9, 8, 35]. Indeed, it was recently shown [9] that there is a universal parameterization of all totally positive matrices with this property, although this parameterization is not always convenient to use. Second, we show that well known composition laws for producing new TP matrices from previous ones also ....

....Whitney; Whitney; and Erdrei [2, p. 215] show that all totally positive Toeplitz matrices can be assembled from the above operations applied to the basic TP Toeplitz matrices mentioned earlier. A row of an upper triangular totally positive Toeplitz matrix is called a P olya frequency sequence [35, 2]. 5. If A is TP, so is any Schur complement of A. Since any minor of the Schur complement is a quotient of minors of A, high accuracy formulas for minors of A yield high accuracy formulas for minors of the Schur complement. 6. Let A(x 1 ; x 2 ; x p ) be a TP matrix, when 0 x 1 x 2 Delta ....

T. Ando. Totally positive matrices. Lin. Alg. Appl., 90:165--219, 1987.

....Let A be an n Theta n nonnegative matrix. It is totally nonnegative (TN) if all of its minors are nonnegative; it is totally positive (TP) if all of its minors are positive; it is oscillatory (OS) if A is TN and A m is TP for some positive integer m. Evidently, TP OS TN : It is known (see [1, 8, 12]) that the inclusions are all strict, and that the closure of TP is TN. Many authors, motivated by theory and applications, have studied TN, OS, and TP matrices (see for example [1, 3, 4, 7, 8, 9, 10, 11, 12, 15, 17] These classes of matrices have a lot of nice properties that resemble those of ....

....(OS) if A is TN and A m is TP for some positive integer m. Evidently, TP OS TN : It is known (see [1, 8, 12] that the inclusions are all strict, and that the closure of TP is TN. Many authors, motivated by theory and applications, have studied TN, OS, and TP matrices (see for example [1, 3, 4, 7, 8, 9, 10, 11, 12, 15, 17]) These classes of matrices have a lot of nice properties that resemble those of positive semi definite Hermitian matrices. For instance, positive semi definite Hermitian matrices have nonnegative eigenvalues and so do TN, OS, and TP matrices. In fact, if a matrix is TP or OS, then it has ....

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T. Ando, Totally positive matrices, Linear Algebra Appl. 90 (1987), 165-219.

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Ando, T., Totally positive matrices, Lin. Alg. and Appl. 90 (1987), 165--219.

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Ando, T., Totally positive matrices, Lin. Alg. and Appl. 90 (1987), 165-- 219.

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T. Ando. Totally positive matrices. Lin. Alg. Appl., 90:165--219, 1987.

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T. Ando. Totally positive matrices. Lin. Alg. Appl., 90:165--219, 1987.

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T. Ando. Totally positive matrices. Linear Algebra Applic. 90 (1987), 165-219.

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