A Survey on Positive Definite Preserving Transformations On

A survey on positive deﬁnite preserving transformations on symmetric matrix space

Hongru Zhao

Abstract. A survey about positive deﬁnite preserving transformation, which includes the representation theorem for linear and and nonlinear transformation. In additionally, there is a characterization theorem of functions preserving positive deﬁniteness for trees.

1. Positive deﬁnite preserving linear transformations on symmetric matrix space

Definition 1.1. The Hadamard product If A = (aij)n×m and B = (bij)n×m, then the Hadamard product of A and B is denoted by A ◦ B and deﬁned by

A ◦ B = (aijbij)

Lemma 1.1. [1] If A is positive semi-deﬁnite of rank r. If every entry in its diagonal is non-zero, then there exist a vector x = (xi)n×1, xi 6= 0, i = 1, 2, ··· , n such that A = xxH + B, where B is a positive semi-deﬁnite matrix of rank r-1.

Lemma 1.2. (The modified Schur product theorem)[1] If A is positive definite, B is positive semi-definite and all entries on its diagonal of B is non-zero, and B = xxH + C, then A ◦ B is positive definite.

Proof. [1] By previous lemma, we know that there exist a vector x = (xi)n×1, H where xi 6= 0, i = 1, 2, ··· , n such that B = xx + C, where C is a positive semi- deﬁnite matrix. Pn H Suppose A = i=1 xixi , where x1, x2, ··· , xn are linear independent. Since

n n ! n X X X (1.1) αi (xi ◦ x) = 0 ⇔ αixi ◦ x = 0 ⇔ αixi = 0 i=1 i=1 i=1 1 2 HONGRU ZHAO the vectors x1 ◦ x, x2 ◦ x,..., xn ◦ x are linear independent. Thus, A ◦ B = A ◦ xxH + A ◦ C n H X H H > A ◦ xx = xixi ◦ xx i=1 n X H = (xi ◦ x) (xi ◦ x) > 0 i=1 The next theorem is a representation theorem for linear transformation which preserving real positive deﬁnite property.

Theorem 1.1. Let T : Pn(R) −→ Pn(R) is a linear transformation preserving positive deﬁniteness and rank T (Eii) = 1, i = 1, 2, ··· , n. Then there exists an invertible matrix W and a positive semi-deﬁnite matrix H, diagH = diagIn, such that for every A ∈ Pn(R) T (A) = W (H ◦ A)W t

2. Positive deﬁnite preserving nonlinear transformations on symmetric matrix space

2.0.1. [2]Hadamard matrix functions. 2 If n functions fij(t) are given for i, j = 1, 2, ··· , n with appropriate domains, and if A = [aij] ∈ Mn, the function f : A → f(A) = [fij (aij)] is called a Hadamard functions to distinguish it from the usual notion of matrix function. If each function fij(t) is polynomial

kij kij −1 (2.1) fij(t) = akij (i, j)t + akij −1(i, j)t + ··· + a0(i, j)

h kij i Then f(A) = [fij (aij)] = akij (i, j)aij + ··· and we can write the function f(t) as a matrix polynomial that formally looks like: m m−1 f(t) = Amt + Am−1t + ··· + A0, m ≡ max {kij; i, j = 1, . . . , n} Then (m) (m−1) (2.2) f(A) ≡ Am ◦ A + Am−1 ◦ A + ··· + A0. Notice that Hadamard multiplication is commutative. If each function fij(t) is an analytic function with radius of convergence RIij > 0, then the Hadamard function f : A → f(A) = [fij (aij)] can be written as a power series ∞ X (k) f(A) = Ak ◦ A k=0 RUNNING TITLE / HEADER 3 where we must assume that |aij| < Rij for i, j = 1, 2, ··· , n. For example, let f(t) = et, then ∞ ∞ X 1 X 1 f(A) = [eaij ] = A(k) = ak k! k! ij k=0 k=0 Theorem 2.1. [2] Let A be positive semidefinite. If all the coefficient matrices Ak in (2.2) are positive semidefinite, then the Hadamard function f(A) defined by (2.2) is positive semidefinite.

Remark 2.1. If one of the Ak is positive definite and A is also positive semidefinite, then f(A) is positive semidefinite. For the positive definite preserving mapping, we have the following fairly strong necessary conditions. Theorem 2.2. Let f(·) be an (n − 1)-times continuously differentiable real valued function on (0, ∞), and suppose that the Hadamard function f(A) = [f(aij)] is positive semidefinite for every positive semidefinite matrix A that has positive (k) entries. Then f (t) > 0 for all t ∈ (0, ∞) and all k = 0, 1, ··· , n − 1. Corollary 2.1. Let 0 < α < n − 2, α not an integer. There is some positive semidefinite matrix A with positive entries such that the Hadamard power A(α) = α [aij] is not positive semidefinite.

Theorem 2.3. Let A = [aij] be a positive semideﬁnite matrix with nonnegative (α) entries. If α > n − 2, then the Hadamard power A is positive semideﬁnite. Furthermore, the lower bound n = 2 is, in general, the best possible.

(α) Remark 2.2. If A = [aij] be a positive deﬁnite matrix, then A is a positive deﬁnite matrix if α = 1, 2, 3, 4, 5, ··· .

3. Further structure of positive deﬁnite preserving nonlinear transformations on symmetric matrix space

All the following theorems comes from [3].

3.0.1. Representation theorem for positive deﬁnite preserving nonlinear transformations.

Theorem 3.1. Let 0 < α 6 ∞ and let f :(−α, α) → R. For every matrix ∗ A = (aij), denote by f [A] the matrix ∗ f (aij) if i 6= j (3.1) (f [A])ij = aij if i = j Then f ∗[A] is positive semideﬁnite for every positive semideﬁnite matrix A with entries in (−α, α) if and only if f(x) = xg(x) where: 1. g is analytic on the disc D(0, α); 2. kgk∞ 6 1; 4 HONGRU ZHAO

3. g is absolutely monotonic on (0, α). The above result does come as a surprise. It formally demonstrates that, ex- cept in trivial cases, no guarantee can be given that applying a function to the off-diagonal elements of a matrix will preserve positive definiteness. There are thus no theoretical safeguards that thresholding procedures used in innumerable applications will maintain positive definiteness. The following theorem characterizes the class of absolutely monotonic function on (0, α).

Theorem 3.2. Let 0 < α 6 ∞. Then the following are equivalent: 1. f is absolutely monotonic on (0, α); 2. f is the restriction to (0, α) of an analytic function on D(0, α) := {z ∈ C : |z| < α} with positive Taylor coefficients, i.e., ∞ X n (3.2) f(x) = anx (x ∈ (0, α)) n=0 for some an > 0. Remark 3.1. Let 0 < α 6 ∞. A function f :(−α, α) → R can be represented as: ∞ X n (3.3) f(x) = anx (−α < x < α) n=0 for some an > 0 if and only if f extends analytically to D(0, α) and is absolutely monotonic on (0, α). 3.0.2. Theorem for positive definite preserving nonlinear transformations in graph. Let G = (V,E) be an undirected graph with n > 1 vertices V = {1, . . . , n} and edge set E. Two vertices a, b ∈ V, a 6= b, are said to be adjacent in G if (a, b) ∈ E. A graph is simple if it is undirected, and does not have multiple edges or self-loops. We say that the graph G0 = (V 0,E0) is a subgraph of G = (V,E), denotes by G0 ⊂ G, if V 0 ⊆ V and E0 ⊂ E. In addition, if G0 ⊂ G and E0 = (V 0 × V 0) ∩ E, we say that G0 is an induced subgraph of G. A graph G is called complete if every pair of vertices are adjacent. A path of length k > 1 from vertex i to j is a finite sequence fo distinct vertices v0 = i, . . . , vk = j in V and edges (v0, v1) ,..., (vk−1, vk) ∈ E. A k-cycle in G is a path of length k − 1 with an additional edge connecting the two end points. A graph G is called connected if for any pair of distinct vertices i, j ∈ V there exists a path between them. A special class of graphs are trees. These are connected graphs on n vertices with exactly n − 1 edges. A tree can also be defined as a connected graph with no cycle of length n > 3, or as a connected graph with a unique path between any two vertices. Graphs provide a useful way to encode patterns of zeros in symmetric matrices + by letting (i, j) ∈ E if and only if aij 6= 0. Denote by Pn the cone of n×n symmetric positive definite matrices, and by P+ the cone of positive definite matrices of any RUNNING TITLE / HEADER 5 dimension. We shall write A > 0 whenever A ∈ P+. We define the cone of positive definite matrices with zeros according to a given graph G with n vertices by + + PG := A ∈ Pn : aij = 0 if (i, j) ∈/ E, i 6= j . Proposition 3.1. Let G = (V,E) be a connected undirected graph and denote by ∆ = ∆(G) the maximum degree of the vertices of G. Assume f : R → R satisfies (3.4) |f(x)| 6 c|x| ∀x ∈ R 1 + + for some 0 6 c < ∆ . Then f ∗ [A] ∈ PG for every A ∈ PG. + Proof. For every A ∈ PG, denote by MA the matrix with entries

 f(aij ) if aij 6= 0 and i 6= j  aij (3.5) (MA)ij = 1 if i = j  0 if aij = 0 and i 6= j The matrix f ∗[A] can be written as ∗ f [A] = A ◦ MA 1 Since 0 6 c < ∆ , an application of Gershgorin’s circle theorem demonstrates that MA > 0. As a consequence, by the Schur product theorem, A ◦ MA > 0 for every + A ∈ PG. The next theorem is a characterization of functions preserving positive deﬁ- niteness for trees. Firstly, we deﬁne a class of functions contracting at the origin, (3.6) C := {f : R → R : |f(x)| 6 |x|∀x ∈ R}. Theorem 3.3. Let G = (V,E) be a graph. Then ∗ + + (3.7) f : R → R : f [A] ∈ PGfor every A ∈ PG = C if and only if G is a tree.

References 1. Huynh Dinh Tuan-Tran Thi Nha Trang-Doan The Hieu (2010). Positive deﬁnite preserving linear transformations on symmetric matrix spaces. 2. Roger A. Horn and Bala Rajaratnam. Topics in matrix analysis. Cambridge University Press, Cambridge, 1991. 3. Dominique Guillot and Charles R. Johnson. POSITIVE MAPS, ABSOLUTELY MONO- TONIC FUNCTIONS AND THE REGULARIZATION OF POSITIVE DEFINITE MATRI- CES

Department of Mathematics, University of Minnesota Duluth, US Email address: [email protected]