Matrix positivity preservers

These are some notes written while visiting IISc Bangalore. This was a discussion session with Apoorva Khare and Sujit Damase. A nice reference to see these things in more detail is Apoorva’s survey [1].

Acknowledgement

Apoorva read this and wrote to me with some corrections and additional comments. I have added some of those comments as footnotes. I express my gratitude to him for taking out the time.

Schoenberg’s theorem and positivity preservers

All dimensions

Take a function \(f: I \rightarrow \mathbb{R}\), for \(I=\) any domain in \(\mathbb{R}\). Let \((a_{ij}) \in \mathbb{P}_{n}(I)\), where \(\mathbb{P}_{n}(I)\) is the set of \(n \times n\) positive semidefinite (real symmetric) matrices with entries in \(I\).

We want to characterize \(f\) such that for all \((a_{ij}) \in \mathbb{P}_{n}(I)\), \((f(a_{ij})) \in \mathbb{P}_{n}(\mathbb{R})\).

Let’s call this family \(\mathcal{F}_{\infty}(I)\), or when the context is clear, simply \(\mathcal{F}_{\infty}\).

\(f=1\) obviously works, and \(f(x)=x\) also works. The class of \(f\) that work is closed under addition, scaling by positive reals and pointwise limits. So it is a closed convex cone. It is also closed under pointwise multiplication because of the Schur product theorem (1911).

Schur showed that matrices preserve PSD condition under Hadamard products (which should be named after Schur actually). So this easily implies \(\mathcal{F}_{\infty}\) being closed under products.

These above remarks can be collected into the following theorem.

(1925) Polya,-Szego’s book exercise

The positivity preservers \(\mathcal{F}_{\infty}\)(I) contain the closure of all polynomials, and therefore contains all power series with non-negative coefficients that are convergent on \(I\).

Are there any other functions? What are they?

The answer is no if you want \(f\) to work for all \(\mathbb{P}_{n}(I)\).

(Schoenberg, 1942)

  1. No, if \(f\) is continuous and \(I=[-1,1]\).

(Rudin, 1959)

  1. No, if \(I=(-1,1)\) and any \(f\). This implies analyticity! In this paper, Rudin also observed that \(\mathcal{F}_{\infty}(\mathbb{C}) \supset \sum_{k,l \geq 0}^{} a_{kl}z^{k} \overline{z}^{l} \in [0,\infty)\) with \(a_{k,l} \geq 0\). He then conjectured that these are all.

(Herz, 1963)

  1. Not beyond Rudin’s conjecture, for \(I \subseteq \mathbb{C}\) the open unit.

Now, let’s go back to \(I=[-1,1]\). In Rudin’s paper, he also noticed that the endpoints might introduce some new functions. For example \(x,x^{3},x^{5},\dots\) converge to \(\mathbf{1}_{1}- \mathbf{1}_{-1}\) where \(\mathbf{1}\) is the indicator function. Let \(\varphi_{-1}(x)= \mathbf{1}_{1} - \mathbf{1}_{-1}\).

Similarly, the even powers \(1,x^{2},x^{4},\dots\) converge to \(\mathbf{1}_{\{-1,1\}} = \varphi_{-2}(x)\). Then, Rudin asked if there are any others?

(Christensen-Ressel, 1978)

\(I=[-1,1]\) for any \(f\). Then \(\mathcal{F}_{\infty}([-1,1]) = \sum_{k \geq -2}^{}a_{k} \varphi_{k}(x)\) and \(a_{k} \geq 0\) and for \(k \geq 0\), \(\varphi_{k}(x) = x^{k}\).

There is also a related work for the closed disc for which one could see Apoorva’s survey [1].

For fixed dimension

Define \[\begin{equation} \mathcal{F}_{n}(I) = \{ \text{ positivity preservers for $n \times n$ matrices with entries in }I\}. \end{equation}\] Then \[\begin{equation} \mathcal{F}_{\infty}(I) = \bigcap_{n \geq 1} \mathcal{F}_{n}(I). \end{equation}\]

  1. \(\mathcal{F}_{1}(I)\) is trivial. It is just all the functions taking \(I \cap \mathbb{R}_{\geq 0}\) to \(\mathbb{R}_{ \geq 0}\).
  2. For \(\mathcal{F}_{2}((0,\infty))\) the problem was solved by Vasudeva. It is the cone of functions satisfying \(f \geq 0\), \(f\) non-decreasing and \(f\) satisfying multiplicative midconvexity: \[\begin{equation} f(x)f(y) \geq f(\sqrt{xy})^{2} \end{equation}\] It’s the same thing as saying \(\log f(e^{x})\) is midconvex. These properties are sufficient to imply continuity.

Open question

What is \(\mathcal{F}_{3}((0,\infty))\)?

Which polynomials work for \(\mathcal{F}_{n}(I)\)? This is partially known. 1

Apoorva has a result where they show that \(\mathcal{F}_{n}(I)\) has polynomials with non-negative coefficients. This is about his work with Terry Tao.

There is also a theorem of Loewner (1967).2

Let \(f \in \mathcal{F}_{n}((0,\infty)) \implies f \in C^{(n-3)}((0,\infty))\) and \(f,f^{1},\dots,f^{(n-3)} \geq 0\).

Motivation

Positivity preservers are used as cutoffs (regularizers) of covariance matrices in statistics. Apoorva explained this using how this is connected to modeling weather. See [1] for further details. These slides go into this aspect in more detail.

Connection to sphere packing

Let \((X_{n},d)\) be a finite metric space with \(n+1\) elements. All the data of \((X_{n},d)\) is inside a distance matrix: \[\begin{equation} D_{X} = (d_{X_{n}}(x_{i},x_{j}))_{i,j=0}^{n} \end{equation}\]

Frechet (1010)

Then there is an isometric embedding \((X_{n},d) \rightarrow (\mathbb{R}^{n}, \| \cdot\|_{\infty})\).

Witsenhausen sharpened it to \(\mathbb{R}^{n-1}\) for \(n \geq 3\).

When does \((X_{n},d) \rightarrow (\mathbb{R}^{m} , \|\cdot\|_{2})\) isometrically? Clearly \(m=n\) should be sufficient. What is the minimum \(m\) given \(X_{n}\)?

(Schoenberg, 1935)

Construct \(\mathrm{CM}(X_{n}) = (d_{i0}^{2} + d_{j0}^{2} - d_{ij}^{2})_{i,j=1}^{n}\). Then \(X_{n}\) embeds in \(\mathbb{R}^{n}\) with \(\|\cdot\|_{2}\)-norm if and only if \(\mathrm{CM}(X_{n})\) is PSD. Also, minimum dimension \(m = \mathrm{rank}(\mathrm{CM}(X_{n}))\).

Let us prove the \(\implies\) implication. Observe that \[\begin{align} d_{i0}^{2} + d_{j0}^{2} - d_{ij}^{2} & = \|x_{i}-x_{0}\|^{2} + \|x_{j}-x_{0}\|^{2} - \|(x_{i}-x_{0}) - (x_{j}- x_{0})\|^{2} \\ & = 2 \langle x_{i} - x_{0}, x_{j} - x_{0}\rangle. \end{align}\] This shows that \(\mathrm{CM}(X_n)\) is a Gram matrix.

The other direction is to basically find \(n\) points such that \(\mathrm{CM}(X)\) is a Gram matrix of those \(n\)-points and map \(X\) to those \(n\)-points.

In the same paper, Schoenberg also dealt with embedding \(X \rightarrow (S^{n}, \angle)\) for the \(n\)-dimensional sphere in \(\mathbb{R}^n\). This takes us to the doorstep of positivity preservers.

(Schoenberg 1935)

There exists an isometric embedding \(X \rightarrow (S^{n}, \angle)\) for the \(n\)-dimensional sphere if and only if

  1. \(\mathrm{diam}(X) \leq \pi\),
  2. \(\cos [D_{X}] \in \mathbb{P}_{n+1}\). That is, cosine is a Schoenberg-positive definite function on \(X\) 3. Here \(D_{X}\) is the distance matrix.

(Schoenberg) What are all the positive definite functions on \(S^{n-1}\)?

He answered this in his 1942 paper. This is the same as asking which functions \(f \circ \cos^{-1}: [-1,1] \rightarrow \mathbb{R}\) send correlation matrices to correlation matrices. This is answered as \[\begin{equation} f = \sum_{k=0}^{\infty} a_{k} C_{k}^{(\cdot)} (x), \text{ with }a_{k} \geq 0, \end{equation}\] and \(C_{k}^{(\cdot)}(x)\) are some Gegenbauer polynomials.

(Schoenberg)

What about positive-definite functions on \(S^{\infty}\)? This is the set of \(l^{2}\)-sequences with \(l^{2}\)-norm 1.

This is answered in Schoenberg’s 1942 paper. The answer is \[\begin{equation} \bigcap_{n \geq 3}\{ \sum_{k \geq 0}^{} a_{k} C_{k}^{(\frac{n-3}{2})}(x)\} = \{ \sum_{k \geq 0}^{}a_{k} x^{k}\}. \end{equation}\] Here \(a_{k}\) are non-negative. This is also the same as answering the question of Polya-Szego.

Connection to LP bound

A p.d. kernel is a function \(K: X \times X \rightarrow \mathbb{C}\) on a finite set if the matrix \(K(x_{i},x_{j})\) is psd. For a group \(G\), the function \(\varphi: G \rightarrow \mathbb{C}\) is psd if \(\varphi(g_{i}^{-1} g_{k})\) is psd.

For a function \(K:G \rightarrow G \rightarrow \mathbb{C}\), \(K(x,y)\) is \(G\)-invariant if \(K(gx,gy)= K(x,y)\) for all \(g \in G\). Then, \(\psi(x)=K(x,e)\) is just a one-variable function on \(G\) that determines the kernel \(K\).

Two point homogeneous space

We conisder a two-point homogeneous space \((X,d)\), that is a metric \(G\)-space such that \[\begin{equation} d(x_{1},y_{1})=d(x_{2},y_{2}) \implies \exists g \in G, gx_{1} = y_{1}, g x_{2}=y_{2}. \end{equation}\]

\(K: X \times X \rightarrow \mathbb{C}\) be a function on some \(G\)-space \(X\), for eg. \(X=S^{n-1} \subseteq \mathbb{R}^{n}\), \(G = \mathrm{SO}(n)\). Then because of \(\mathrm{SO}(n-1)\)-invariance, \(K\) becomes a function of \(\langle x,y \rangle\). Hence, \(K = \varphi(\langle x,y\rangle)\) for a one-variable function \(\varphi: [-1,1] \rightarrow \mathbb{C}\). Then, \(K\) is a positive-definite kernel if and only if \(\varphi\) is a positivity preserver.

Bochner’s theorem

Let’s take \(G=S^{1}\), \(f: S^{1} \rightarrow \mathbb{C}\). The theorem is that

(Bochner)

Let \(G\) be a locally compact Abelian group. Then \(f: G \rightarrow \mathbb{C}\) is a continuous positive definite function if and only if there exists some \(\mu \in M(\widehat{G})\)=finite, positive measures on \(\widehat{G}\), such that \[\begin{equation} f = \int_{\widehat{G}}^{} \chi(t) \, \mathrm{d}\mu (\chi) \end{equation}\]

For example, if \(G = S^{1}\), \(\widehat{G} = \mathbb{Z}\) then the positive definite functions on \(G\) are \[\begin{equation} f(t) = \sum_{n \in \mathbb{Z}}^{} e^{i nt} \mu_{n}, \end{equation}\] for some \(l^{1}\)-sequence \(\{\mu_{n}\}_{n \in \mathbb{Z}}\). These essentially look like power series with non-negative coefficients.

For a general sphere, \[\begin{equation} L^{2}(X) = \bigoplus_{j} V_{j}. \end{equation}\] This is a subspace of \(L^{2}(G)\), the \(\mathrm{SO}(n-1)\)-invariant functions of \(\mathrm{SO}(n)\). The Gegenbauer polynomials form an orthonormal basis for \(V_{k}\). Hence, we can write a version of Bochner’s theorem as \[\begin{equation} f(\langle x,y\rangle) = K(x,y) = \sum_{k=0}^{\infty} a_{k} C_{k}^{(\frac{n-3}{2})}, \end{equation}\] where \[\begin{equation} a_{k} = \int_{S^{n-1}}^{} K(x,y) C_{k}^{(\frac{n-3}{2}) }(x,y) \, \mathrm{d}\nu (x). \end{equation}\]

References

1.
A. Khare, The entrywise calculus and dimension-free positivity preservers , with an appendix on sphere packings. arXiv preprint arXiv:2511.08406, (2025).

  1. Apoorva had some comments on this, which I have reproduced below:
    - By Polya-Szego 1925, any polynomial with nonnegative coefficients preserves positivity in all dimensions, when applied entry-wise.
    - Now if you want entry-wise preservers in a fixed dimension, well, they have to preserve psd on a far smaller set of test matrices (only of a single dimension), so one would expect lots more preservers. Even among polynomials!
    - But from 1925 to 2016 there was no example known, of a dimension \(n>2\) and a polynomial preserver in dimension \(n\), which has a negative coefficient.
    - In 2016, we worked out the first such examples with Belton-Guillot-Putinar in a restricted setting, and then with Tao in far more generality. (Also for sums of real powers, not just for polynomials.)
    - For dimension \(n\), what we have worked out fully is, all polynomials with up to \(n+1\) monomials, which preserve positivity on matrices with positive entries.↩︎

  2. Apoorva adds: modulo small tweaks, it remains essentially the only necessary condition for a general function \(f\) and an integer \(n>2\) to date!↩︎

  3. Apoorva adds: In case you want, it may be worth pointing out the distinction between this notion of positive definite functions, and the classical, “Bochner” notion, which has domain pairs of points in a group or a homogeneous space. In case you want to see the connection, please see e.g. Section 3.3 in my survey.↩︎

This page was updated on February 5, 2026.
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