Totally Positive Matrices T. Ando* Division of Applied Mathematics Research Znstitute of Applied Electricity Hokkaido University Sappmo, Japan Submitted by George P. Barker ABSTRACT Though total positivity appears in various branches of mathematics, it is rather unfamiliar even to linear algebraists, when compared with positivity. With some unified methods we present a concise survey on totally positive matrices and related topics. INTRODUCTION This paper is based on the short lecture, delivered at Hokkaido Univer- sity, as a complement to the earlier one, Ando (1986). The importance of positivity for matrices is now widely recognized even outside the mathematical community. For instance, positive matrices play a decisive role in theoretical economics. On the contrary, total positivity is not very familiar even to linear algebraists, though this concept has strong power in various branches of mathematics. This work is planned as an invitation to total positivity as a chapter of the theory of linear and multilinear algebra. The theory of totally positive matrices originated from the pioneering work of Gantmacher and Krein (1937) and was brought together in their monograph (1960). On the other hand, under the influence of I. Schoenberg, Karlin published the monumental monograph on total positivity, Karlin (1968), which mostly concerns totally positive kernels but also treats the discrete version, totally positive matrices. Most of the materials of the present paper is taken from these two monographs, but some recent contributions are also incorporated. The novelty *The research in this paper was supported by a Grant-in-Aid for Scientific Research. LINEAR ALGEBRA AND ITS APPLICATIONS 90:165-219 (1987) 165 0 Elsevier Science Publishing Co., Inc., 1987 52 Vanderbilt Ave., New York, NY 19917 9024-3795/87/$3.50 166 T. AND0 is in the systematic use of skew-symmetric products of vectors and Schur complements of matrices as the key tools to derive the results in a transparent way. The paper is divided into seven sections. In Section 1 classical de- terminantal identities are proved for later use. The notions of total positivity and sign regularity are introduced in Section 2, and effective criteria for total positivity are presented. Section 3 is devoted to the study of various methods of production of new totally positive matrices from given ones. In Section 4 a simple criterion for a totally positive matrix to have a strictly totally positive power is given. Section 5 is devoted to the study of the relationship between the sign regularity of a matrix and the variation-diminishing property of the linear map it induces. In Section 6 the refined spectral theorems of Perron- Frobenius type are established for totally positive matrices. Examples of totally positive matrices are collected in Section 7. But the most significant results, concerning the total positivity of Toeplitz matrices and translation kernels, are only mentioned without proof. 1. DETERMINANTAL IDENTITIES This section is devoted to the derivation of classical spectral and de- terminantal identities, which are used in the subsequent sections. The use of skew-symmetric products of vectors and Schur complements of matrices will unify and simplify the proofs. For each n 2 1, let %n stand for the (real or complex) linear spaces of (column) n-vectors x’= (xi), equipped with inner product (2, y’) := i x&. (1.1) i=l The canonical orthononnal basis of sn consists of the vectors < ( = <(“‘), i = 1,2 ,...> n, with 1 as its i th component, 0 otherwise. A vector x’= (xi) is positive (respectively, strictly positive), in symbols x’> 0 ( > 0), if xi > 0 (>O)fori=1,2 ,..., n. A linear map from &, to 2” is identified with its n x m matrix A = [aij], relative to the canonical basis of .?‘$, and 2”: a. = (As”), eTn)), i=1,2 ,..., n, j = I,2 ,..., m. ‘I I (1.2) The linear map A is also identified with the ordered m-tuple of n-vectors: TOTALLY POSITIVE MATRICES 167 A = [Zl,Zz ,..., a’,], where zi = AeJ”‘), j = 1,2 ,.‘., m. (1.3) A is called positive (strictly positive), in symbols A > 0 (A B 0), if it transforms every nonzero positive vector to a positive (strictly positive) vector. Obviously A is positive (strictly positive) if and only if Zi > 0 ( B 0), i = 1,2,..., m, or equivalently, if and only if a ij > 0 ( > 0), i = 1,2,. , n, j=1,2 ,..., m. k For each k > 1, let @ X” denote the k-tensor space over 2,. The inner k product in @ X,, is determined by k The canonical orthonormal basis of @ Zn is by definition { $,“,“‘@3%“) @ . @qG: 1 G ii G R, j = 1,2 ,..., k}. k Each linear map A from 2, to 2” induces a linear map from @ .%‘,, k k to @ s”, called the k-tensor power and denoted by 8 A: *-- e~k)=(Ax’,)@(Alc’,)8 .*. @(Ax’,). (1.5) If B is a linear map from Xl to Zm, then it follows from (1.5) that (1.6) Let Sk denote the symmetric group of degree k, that is, the group of all pekrmutations of { 1,2,. , k}. Each n E S, gives rise to a linear map I’,‘“) of @ .%“, determined by 168 T. AND0 A k-tensor 2 is called skew-symmetric if PC”)2n = sgn 7r.2 for any 7r E S,, (1.8) where sgn r = 1 or - 1 according as T is an even or odd permutation. The subspace of all skew-symmetric k-tensors over Xn is called the kth skew- k symmetric (or kth Grassmunn) space over Xn, and denoted by AZn. The orthogonal projection Pi”’ to i\& is given by Pi”) = & C sgnr. P,‘“). (1.9) n E S, The k-tensor is called the kth skew-symmetric product of the ordered k-tuple { 21, 22,. , Sk}. Then it foil ows from (1.8) and (1.10) that ?~~~(r) A Zr-~(a) A . A iTm-lckj = sgnmex,- A s2 A . A Sk. (1.11) Further, it follows from (1.4) via the definition of determinant, that (iTIAZ~A... Ai?k,ijlAy’,A ... A gkk> = & det[(zi, gj>], (1.12) where det means determinant. A consequence is that { ?‘i, ;a,. , ;k} is linearly dependent if and only if ?r A z2 A . A zk = 0. It follows from (1.5) ;nd (1.7) that, for each linear map A from 2m to Xn, its k-tensor power @ A intertwines P,‘“) and P,‘“) in the sense that PJ”).( &A) =( i A)-Pd”‘) for rr~S,. (1.13) k Therefore @ A intertwines the projections Pi”) and Pi’“‘: Pi”).( &A)=( &A).Pi’? (1.14) TOTALLY POSITIVE MATRICES 169 k The restriction of @ A to the skew-symmetric space is called the k-exterior k power of A, and denoted by AA. I n view of (1.5), the exterior power l\A is determined by the formula A?k)=(Ai?l)A(A?2)A ... A(Ask). (1.15) If I, stands for the identity map of %,,, then h”= ZhSI”,Y (1.16) k the right-hand side being the identity map of A.%?,,. It follows from (1.5) or (1.15) that if B is a linear map from Sr to Xm, I\= (;\A)-( iZ3). (1.17) A conyquence of (1.16) and (1.17) is that if A is an invertible map of Xn, then l\A is invertible, and (1.18) When l<k<n, Qk,* will denote the totality of strictly increasing sequences of k integers chosen from { 1,2,. , n } : Q k,n E a= (d The order relation (YQ fi for (Y,p E Qk, n means by definition that (Y~< pi, i = 1,2,. , k. The complement a’ is the increasingly rearranged {1,2 ,..., n} \ a, so that 0~’ is an element of Qn_k,n. When (YE Qk,*, P E Q, n, and (Yn j3= 0, their union (YU p should be always rearranged increasingly to become an element of Qk+l,n. 170 T. AND0 For each (YE Qk, “, its dispersion number d(a) is defined by k-l d(a):= c (q+i-ei-l)=cxk-(hi-(k-l), (1.20) i=l with the convention d(a) = 0 for (YE Qi, n. Then d(a) = 0 means that (Y consists of k consecutive integers. For (YE Qk, ,, the a-projection of an n-vector x’= (xi) is the k-vector with components xa,, x,*, . , xak. The space of all a-projections is denoted by ;x&, that is, Xa is -r;” indexed by ( (yi>as,. * >a,&. Let A be an n X m matrix, (YE Qk,n, and /I E QI,m. Then A[a]p] is by definition the k X 1 submatrix of A using rows numbered by (Y and columns numbered by p. If A is considered a linear map from ,ri”, to %$, then A[ a]P] is one from SD to Za. When 01= fi, A[a]a] is simply denoted by A [ a]. Further we shall use the following notation: AblP):=A[W], A(@] := Ab’lP], A(alb):=A[4P’], A(a) := A[a’]e’], and A[-(Pl:=A[l,2,...,nlP], A[+] :=A[al1,2 ,..., m], A[-(/3) := A[1,2 ,..., nib’], A(cy]-] := A[a’(1,2 ,..., m]. Given (YE Qk+, let us use the abbreviation q ( = e(“); ) := $;) A G;’ A . * &G+$‘. (1.21) Then by (1.12), {fi!Z: : a E Qk, n } becomes a complete orthonormal system of the k-skew-symmeitic space over &, and is taken as the canonical orthonormal basis of AXn. Therefore the notions of positivity for a k-skew- symmetric tensor and a linear map between skew-symmetric spaces always refer to these canonical basis.