# On the Geometry of the Set of Symmetric Matrices with Repeated Eigenvalues

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## Abstract

We investigate some geometric properties of the real algebraic variety \(\Delta \) of symmetric matrices with repeated eigenvalues. We explicitly compute the volume of its intersection with the sphere and prove a Eckart–Young–Mirsky-type theorem for the distance function from a generic matrix to points in \(\Delta \). We exhibit connections of our study to real algebraic geometry (computing the Euclidean distance degree of \(\Delta \)) and random matrix theory.

## Keywords

Integral geometry Random matrices Euclidean distance degree theory## 1 Introduction

*discriminant*) of real symmetric matrices with repeated eigenvalues and of unit Frobenius norm:

*Q*, the dimension of the space of symmetric matrices is \(N :=\frac{n(n+1)}{2}\) and \(S^{N-1}\) denotes the unit sphere in \(\text {Sym}(n,\mathbb {R})\) endowed with the Frobenius norm \(\Vert Q\Vert :=\sqrt{\text {tr}(Q^2)}\).

This discriminant is a fundamental object and it appears in several areas of mathematics, from mathematical physics to real algebraic geometry, see for instance (Arnold 1972, 1995, 2003, 2011; Teytel 1999; Agrachev 2011; Agrachev and Lerario 2012; Vassiliev 2003). We discover some new properties of this object (Theorems 1.1, 1.4) and exhibit connections and applications of these properties to random matrix theory (Sect. 1.4) and real algebraic geometry (Sect. 1.3).

*discriminant polynomial*:

*Q*. Moreover, it is a sum of squares of real polynomials (Ilyushechkin 2005; Parlett 2002) and \(\Delta \) is of codimension two. The set \(\Delta _\text {sm}\) of smooth points of \(\Delta \) is the set of real points of the smooth part of the Zariski closure of \(\Delta \) in \(\text {Sym}(n,\mathbb {C})\) and it consists of matrices with

*exactly*two repeated eigenvalues. In fact, \(\Delta \) is stratified according to the multiplicity sequence of the eigenvalues; see (1.3).

### 1.1 The Volume of the Set of Symmetric Matrices with Repeated Eigenvalues

Our first main result concerns the computation of the *volume*\(|\Delta |\) of the discriminant, which is defined to be the Riemannian volume of the smooth manifold \(\Delta _{\text {sm}}\) endowed with the Riemannian metric induced by the inclusion \(\Delta _\text {sm}\subset S^{N-1}\).

### Theorem 1.1

### Remark 1

*condition number*of linear problems (Demmel 1988). The very first result gives the volume of the set \(\Sigma \subset \mathbb {R}^{n^2}\) of square matrices with zero determinant and Frobenius norm one; this was computed in Edelman and Kostlan (1995) and Edelman et al. (1994):

*symmetric*matrices with determinant zero in Lerario and Lundberg (2016), with similar expressions. Recently, in Beltrán and Kozhasov (2018) the above formula and Lerario and Lundberg (2016, Thm. 3) were used to compute the expected condition number of

*the polynomial eigenvalue problem*whose input matrices are taken to be random.

In a related paper Breiding et al. (2017) we use Theorem 1.1 for counting the average number of singularities of a random spectrahedron. Moreover, the proof of Theorem 1.1 requires the evaluation of the expectation of the square of the characteristic polynomial of a GOE(*n*) matrix (Theorem 1.6 below), which constitutes a result of independent interest.

*average*number of symmetric matrices with repeated eigenvalues in a uniformly distributed projective two-plane \(L \subset \mathrm {P}\text {Sym}(n, \mathbb {R})\simeq \mathbb {R}\text {P}^{N-1}\):

### Remark 2

Consequence (1.1) combined with (1.2) “violates” a frequent phenomenon in random algebraic geometry, which goes under the name of *square root law*: for a large class of models of random systems, often related to the so called Edelman–Kostlan–Shub–Smale models (Edelman and Kostlan 1995; Shub and Smale 1993b, a, c; Edelman et al. 1994; Kostlan 2002), the average number of solutions equals (or is comparable to) the square root of the maximum number; here this is not the case. We also observe that, surprisingly enough, the *average cut* of the discriminant is an integer number (there is no reason to even expect that it should be a rational number!).

More generally one can ask about the expected number of matrices with a multiple eigenvalue in a “random” compact 2-dimensional family. We prove the following.

### Theorem 1.2

- 1.
with probability one the map \(\pi \circ F\) is an embedding and

- 2.
the expected number of solutions of the random system \(\{f_1=f_2=0\}\) is finite.

### Example 1

When each \(f_i\) is a Kostlan polynomial of degree *d*, then the hypotheses of Theorem 1.2 are verified and \(\mathop {\mathbb {E}}\limits \#\{f_1=f_2=0\}=2d |\Omega |/|S^2|\); when each \(f_i\) is a degree-one Kostlan polynomial and \(\Omega =S^2\), then \(\mathop {\mathbb {E}}\limits \#\{f_1=f_2=0\}=2\) and we recover (1.1).

### 1.2 An Eckart–Young–Mirsky-Type Theorem

The classical Eckart–Young–Mirsky theorem allows to find a best low rank approximation to a given matrix.

*r*. Then for a given \(m\times n\) real or complex matrix

*A*a rank

*r*matrix \({\tilde{A}}\in \Sigma _r\) which is a global minimizer of the distance function

*a best rank r approximation to A*. The Eckart–Young–Mirsky theorem states that if \(A=U^*SV\) is the singular value decomposition of

*A*, i.e.,

*U*is an \(m\times m\) real or complex unitary matrix,

*S*is an \(m\times n\) rectangular diagonal matrix with non-negative diagonal entries \(s_1\ge \cdots \ge s_m\ge 0\) and

*V*is an \(n\times n\) real or complex unitary matrix, then \({\tilde{A}} = U^*{\tilde{S}}V\) is a best rank

*r*approximation to

*A*, where \({\tilde{S}}\) denotes the rectangular diagonal matrix with \({\tilde{S}}_{ii} = s_i\) for \(i=1,\ldots ,r\) and \({\tilde{S}}_{jj}=0\) for \(j=r+1,\ldots ,m\). Moreover, a best rank

*r*approximation to a sufficiently generic matrix is actually unique. More generally, one can show that any critical point of the distance function \(\text {dist}_A:\Sigma _r \rightarrow \mathbb {R}\) is of the form \(U^*{\tilde{S}}^I V,\) where \(I\subset \{1,2,\ldots ,m\}\) is a subset of size

*r*and \({\tilde{S}}^I\) is the rectangular diagonal matrix with \({\tilde{S}}^I_{ii} = s_i\) for \(i\in I\) and \({\tilde{S}}^I_{jj} = 0\) for \(j\notin I\). In particular, the number of critical points of \(\text {dist}_A\) for a generic matrix

*A*is \(\genfrac(){0.0pt}1{n}{r}\). In Draisma et al. (2016) the authors call this count the

*Euclidean distance degree*of \(\Sigma _r\); see also Sect. 1.3 below.

In the case of real symmetric matrices similar results are obtained by replacing singular values \(\sigma _1\ge \cdots \ge \sigma _n\) with absolute values of eigenvalues \(|\lambda _1|>\cdots >|\lambda _n|\) and singular value decomposition \(U\Sigma V^*\) with spectral decomposition \(C^T\Lambda C\); see Helmke and Shayman (1995, Thm. 2.2) and Lerario and Lundberg (2016, Sec. 2).

For the distance function from a symmetric matrix to the cone over \(\Delta \) we also have an Eckart–Young–Mirsky-type theorem. We prove this theorem in Sect. 2.

### Theorem 1.3

### Remark 3

Since \(\mathscr {C}(\Delta )\subset \text {Sym}(n,\mathbb {R})\) is the homogeneous cone over \(\Delta \subset S^{N-1}\) the above theorem readily implies an analogous result for the spherical distance function from \(A\in S^{N-1}\) to \(\Delta \). The critical points are \((1-\frac{(\lambda _i-\lambda _j)^2}{2})^{-1/2}\,C^T \Lambda _{i,j} C\) and the global minimum of the spherical distance function \(\text {d}^S\) is \(\min _{B\in \Delta } \text {d}^S(A,B) = \min _{1\le i<j\le n} \arcsin \left( \tfrac{|\lambda _i-\lambda _j|}{\sqrt{2}}\right) \).

*stratum*of \(\mathscr {C}(\Delta )\). These strata are in bijection with vectors of natural numbers \(w=(w_1,w_2,\ldots ,w_n)\in \mathbb {N}^n\) such that \(\sum _{i=1}^n i \,w_i = n\) as follows: let us denote by \(\mathscr {C}(\Delta )^w\) the smooth semialgebraic submanifold of \(\text {Sym}(n,\mathbb {R})\) consisting of symmetric matrices that for each \(i\ge 1\) have exactly \(w_i\) eigenvalues of multiplicity

*i*. Then, by Shapiro and Vainshtein (1995, Lemma 1), the semialgebraic sets \(\mathscr {C}(\Delta )^w\) with \(w_1<n\) form a stratification of \(\mathscr {C}(\Delta )\):

### Theorem 1.4

- 1.
Any critical point of the distance function \(\hbox {d}_A: \mathscr {C}(\Delta )^w \rightarrow \mathbb {R}\) is of the form \(C^T \tilde{\Lambda } C\), where \(\tilde{\Lambda }\in \hbox {Diag}(n,\mathbb {R})^w\) is the orthogonal projection of \(\Lambda \) onto one of the planes in \(\overline{\hbox {Diag}(n,\mathbb {R})^w}\).

- 2.
The distance function \(\hbox {d}_A: \mathscr {C}(\Delta )^w \rightarrow \mathbb {R}\) has exactly \(\frac{n!}{1!^{w_1}2!^{w_2}3!^{w_3}\cdots }\) critical points, one of which is the unique global minimum of \(\hbox {d}_A\).

### Remark 4

Note that the manifold \(\mathscr {C}(\Delta )^w\) is not compact and therefore the function \(\hbox {d}_A: \mathscr {C}(\Delta )^w \rightarrow \mathbb {R}\) might not a priori have a minimum.

### 1.3 Euclidean Distance Degree

Let \(X\subset \mathbb {R}^m\) be a real algebraic variety and let \(X^\mathbb {C}\subset \mathbb {C}^m\) denote its Zariski closure. The number \(\#\{x\in X_{\text {sm}}: u-x\perp T_x X_{\text {sm}}\}\) of critical points of the distance to the smooth locus \(X_{\text {sm}}\) of *X* from a generic point \(u\in \mathbb {R}^m\) can be estimated by the number \(\text {EDdeg}(X):=\#\{x\in X_{\text {sm}}^{\mathbb {C}} : u-x \perp T_x X_{\text {sm}}^{\mathbb {C}}\}\) of “complex critical points”. Here, \(v\perp w\) is orthogonality with respect to the bilinear form \((v,w)\mapsto v^Tw\). The quantity \(\text {EDdeg}(X)\) does not depend on the choice of the generic point \(u\in \mathbb {R}^m\) and it’s called *the Euclidean distance degree* of *X* (Draisma et al. 2016). Also, solutions \(x\in X^{\mathbb {C}}_{\text {sm}}\) to \(u-x\perp T_xX_{\text {sm}}^{\mathbb {C}}\) are called *ED critical points of u with respect to X* (Drusvyatskiy et al. 2017). In the following theorem we compute the Euclidean distance degree of the variety \(\mathscr {C}(\Delta )\subset \text {Sym}(n,\mathbb {R})\) and show that all ED critical points are actually real (this result is an analogue of Drusvyatskiy et al. (2017, Cor. 5.1) for the space of symmetric matrices and the variety \(\mathscr {C}(\Delta )\)).

### Theorem 1.5

Let \(A\in \text {Sym}(n,\mathbb {R})\) be a sufficiently generic symmetric matrix. Then the \({n \atopwithdelims ()2}\) real critical points of \(\text {d}_A: \mathscr {C}(\Delta _{\text {sm}}) \rightarrow \mathbb {R}\) from Theorem 1.3 are the only ED critical points of *A* with respect to \(\mathscr {C}(\Delta )\) and the Euclidean distance degree of \(\mathscr {C}(\Delta )\) equals \(\text {EDdeg}(\mathscr {C}(\Delta )) = {n \atopwithdelims ()2}\).

### Remark 5

An analogous result holds for the closure of any other stratum of \(\mathscr {C}(\Delta )^w\). Namely, \(\text {EDdeg}(\overline{\mathscr {C}(\Delta )^w})=\frac{n!}{1!^{w_1}2!^{w_2}3!^{w_3}\cdots }\) and for a generic real symmetric matrix \(A\in \text {Sym}(n,\mathbb {R})\) ED critical points are real and given in Theorem 1.4.

### 1.4 Random Matrix Theory

The proof of Theorem 1.1 eventually arrives at Eq. (3.7), which reduces our study to the evaluation of a special integral over the *Gaussian Orthogonal Ensemble* (\(\mathrm {GOE}\)) (Mehta 2004; Tao 2012). The connection between the volume of \(\Delta \) and random symmetric matrices comes from the fact that, in a sense, the geometry in the Euclidean space of symmetric matrices with the Frobenius norm and the random \(\mathrm {GOE}\) matrix model can be seen as the same object under two different points of view.

The integral in (3.7) is the second moment of the characteristic polynomial of a \(\mathrm {GOE}\) matrix. In Mehta (2004) Mehta gives a general formula for all moments of the characteristic polynomial of a \(\mathrm {GOE}\) matrix. However, we were unable to locate an exact evaluation of the formula for the second moment in the literature. For this reason we added Proposition 4.2, in which we compute the second moment, to this article. We use it in Sect. 4 to prove the following theorem.

### Theorem 1.6

*k*we have

*non-asymptotic*(as opposed to studies in the limit \(n\rightarrow \infty \), Ben Arous and Bourgade 2013; Nguyen et al. 2017) result in random matrix theory. It would be interesting to provide an estimate of the implied constant in (1.4), however this might be difficult using our approach as it probably involves estimating higher curvature integrals of \(\Delta \).

## 2 Critical Points of the Distance to the Discriminant

In this section we prove Theorems 1.3, 1.4 and 1.5. Since Theorem 1.3 is a special case of Theorem 1.4, we start by proving the latter.

### 2.1 Proof of Theorem 1.4

Let us denote by \(\overline{\mathscr {C}(\Delta )^w}\subset \text {Sym}(n,\mathbb {R})\) the Euclidean closure of \(\mathscr {C}(\Delta )^w\). Note that \(\overline{\mathscr {C}(\Delta )^w}\) is a (real) algebraic variety, the smooth locus of \(\overline{\mathscr {C}(\Delta )^w}\) is \(\mathscr {C}(\Delta )^w\) and the boundary \(\overline{\mathscr {C}(\Delta )^w}{\setminus } \mathscr {C}(\Delta )^w\) is a union of some strata \(\mathscr {C}(\Delta )^{w^\prime }\) of greater codimension.

The following result is an adaptation of Bik and Draisma (2017, Thm. 3) to the space \(\text {Sym}(n,\mathbb {R})\) of real symmetric matrices, its subspace \(\text {Diag}(n,\mathbb {R})\) of diagonal matrices and the action \(C\in O(n), A\in \text {Sym}(n,\mathbb {R})\mapsto C^TAC\in \text {Sym}(n,\mathbb {R})\). Note that the assumptions of Bik and Draisma (2017, Thm. 3) are satisfied in this case as shown in Bik and Draisma (2017, Sec. 3).

### Lemma 2.1

Let \(X\subset \text {Sym}(n,\mathbb {R})\) be a *O*(*n*)-invariant subvariety. Then for a sufficiently generic \(\Lambda \in \text {Diag}(n,\mathbb {R})\) the set of critical points of the distance \(\text {d}_{\Lambda }: X_{\text {sm}} \rightarrow \mathbb {R}\) is contained in \(X\cap \text {Diag}(n,\mathbb {R})\).

Let now \(X=\overline{\mathscr {C}(\Delta )^w}\), let \(A\in \text {Sym}(n,\mathbb {R})\) be a sufficiently generic symmetric matrix and fix its spectral decomposition \(A=C^T\Lambda C\). By Lemma 2.1 critical points of \(\text {d}_{\Lambda }: \mathscr {C}(\Delta )^w \rightarrow \mathbb {R}\) are all diagonal. Since the intersection \(X\cap \text {Diag}(n,\mathbb {R})=\overline{\text {Diag}(n,\mathbb {R})^w}\) is an arrangement of \(\frac{n!}{1!^{w_1}2!^{w_2}3!^{w_3}\cdots }\) planes, critical points of \(\text {d}_{\Lambda }\) are the orthogonal projections of \(\Lambda \) on each of the components of the plane arrangement. Moreover, one of these points is the (unique) closest point on \(\overline{\text {Diag}(n,\mathbb {R})^w}\) to \(\Lambda \). The critical points of the distance \(\text {d}_A: \mathscr {C}(\Delta )^w \rightarrow \mathbb {R}\) from \(A=C^T\Lambda C\) are obtained via conjugation of critical points of \(\text {d}_\Lambda \) by \(C\in O(n)\). Both claims follow. \(\square \)

### 2.2 Proof of Theorem 1.3

### 2.3 Proof of Theorem 1.5

In the proof of Theorem 1.3 we showed that there are \(\genfrac(){0.0pt}1{n}{2}\) real ED critical points of the distance function from a general real symmetric matrix *A* to \(\mathscr {C}(\Delta )\). In this subsection we in particular argue that there are no other (complex) ED critical points in this case. The argument is based on Main Theorem from Bik and Draisma (2017) which is stated first.

### Theorem 2.2

(Main Theorem from Bik and Draisma (2017)). Let *V* be a finite-dimensional complex vector space equipped with a non-degenerate symmetric bilinear form, let \(G^{\mathbb {C}}\) be a complex algebraic group and let \(G^{\mathbb {C}}\rightarrow O(V)\) be an orthogonal representation. Suppose that \(V_0\subset V\) is a linear subspace such that, for sufficiently generic \(v_0\in V_0\), the space *V* is the orthogonal direct sum of \(V_0\) and the tangent space \(T_{v_0}G^{\mathbb {C}}v_0\) at \(v_0\) to its \(G^{\mathbb {C}}\)-orbit. Let \(X^\mathbb {C}\) be a \(G^{\mathbb {C}}\)-invariant closed subvariety of *V*. Set \(X^\mathbb {C}_0=X^\mathbb {C}\cap V_0\) and suppose that \(G^{\mathbb {C}}X^\mathbb {C}_0\) is dense in \(X^\mathbb {C}\). Then the \(\text {ED}\) degree of \(X^\mathbb {C}\) in *V* equals the \(\text {ED}\) degree of \(X^\mathbb {C}_0\) in \(V_0\).

We will apply this theorem to the space of complex symmetric matrices \(V=\text {Sym}(n,\mathbb {C})\) endowed with the complexified Frobenius inner product, the subspace of complex diagonal matrices \(V_0=\text {Diag}(n,\mathbb {C})\), the complex orthogonal group \(G^{\mathbb {C}} = \{C\in M(n,\mathbb {C}): C^TC = {\mathbb {1}}\}\) acting on *V* via conjugation and the Zariski closure \(X^\mathbb {C}\subset \text {Sym}(n,\mathbb {C})\) of \(\mathscr {C}(\Delta )\subset \text {Sym}(n,\mathbb {R})\).

*G*-invariant, by Drusvyatskiy et al. (2017, Lemma 2.1), the complex variety \(X^\mathbb {C}\subset \text {Sym}(n,\mathbb {C})\) is also

*G*-invariant. Using the same argument as in Drusvyatskiy et al. (2017, Thm. 2.2) we now show that \(X^\mathbb {C}\) is actually \(G^{\mathbb {C}}\)-invariant. Indeed, for a fixed point \(A\in X^\mathbb {C}\) the map

*G*we must have \(\gamma _A^{-1}(X^\mathbb {C}) = G^{\mathbb {C}}\).

The proof of the statement in Remark 5 is similar. Each plane in the plane arrangement \(\overline{\text {Diag}(n,\mathbb {R})^w}\) yields one critical point and there are \(\frac{n!}{1!^{w_1}2!^{w_2}3!^{w_3}\cdots }\) many such planes. \(\square \)

## 3 The Volume of the Discriminant

The goal of this section is to prove Theorems 1.1 and 1.2. As was mentioned in the introduction, we reduce the computation of the volume to an integral over the \(\mathrm {GOE}\)-ensemble. This is why, before starting the proof, in the next subsection we recall some preliminary concepts and facts from random matrix theory that will be used in the sequel.

### 3.1 The \(\text {GOE}(n)\) Model for Random Matrices

The material we present here is from Mehta (2004).

*probability measure*of any Lebesgue measurable subset \(U\subset \text {Sym}(n,\mathbb {R})\) is defined as follows:

*A*is given by the measure \( \tfrac{1}{Z_n} \int _V e^{-\frac{\Vert \lambda \Vert ^2}{2}} |\Delta (\lambda )|\, d\lambda ,\) where \(d\lambda =\prod _{i=1}^n d\lambda _i\) is the Lebesgue measure on \(\mathbb {R}^n\), \(V\subset \mathbb {R}^n\) is a measurable subset, \(\Vert \lambda \Vert ^2 = \lambda _1^2+\cdots +\lambda _n^2\) is the Euclidean norm, \(\Delta (\lambda ):=\prod _{1\le i<j\le n}(\lambda _j-\lambda _i)\) is the Vandermonde determinant and \(Z_n\) is the normalization constant whose value is given by the formula

### 3.2 Proof of Theorem 1.1

*O*(

*n*) with the left-invariant metric defined on the Lie algebra \(T_{{\mathbb {1}}}O(n)\) by

*O*(

*n*) can be found in Muirhead (1982, Corollary 2.1.16):

*p*is a submersion. Applying to it the smooth coarea formula (see, e.g., Bürgisser and Cucker 2013, Theorem 17.8) we have

*normal Jacobian*of

*p*at \((C,\mu )\) and we compute its value in the following lemma.

### Lemma 3.1

### Proof

Recall that for a smooth submersion \(f: M\rightarrow N\) between two Riemannian manifolds the normal Jacobian of *f* at \(x\in M\) is the absolute value of the determinant of the restriction of the differential \(D_{x}f: T_x M \rightarrow T_{f(x)}N\) of *f* at *x* to the orthogonal complement of its kernel. We now show that the parametrization \(p: O(n)\times (S^{n-2})_* \rightarrow \Delta _{\text {sm}}\) is a submersion and compute its normal Jacobian.

*p*is equivariant with respect to the right action of

*O*(

*n*) on itself and its action on \(\Delta _{\text {sm}}\) via conjugation, i.e., for all \(C, \tilde{C}\in O(n)\) and \(\mu \in S^{n-2}\) we have \(p(C\tilde{C},\mu ) = \tilde{C}^Tp(C,\mu )\tilde{C}\). Therefore, \(D_{(C,\mu )}p = C^TD_{({\mathbb {1}},\mu )}p\, C\) and, consequently, \(\mathrm {NJ}_{(C,\mu )}p = \mathrm {NJ}_{({\mathbb {1}},\mu )}p\). We compute the latter. The differential of

*p*at \(({\mathbb {1}},\mu )\) is the map

*i*,

*j*) where it equals 1. Then \(\{E_{i,j} - E_{j,i} : 1\le i < j\le n\}\) is an orthonormal basis for \(T_{{\mathbb {1}}}O(n)\). One verifies that

*p*is a submersion and

We now compute the volume of the fiber \(p^{-1}(A), A\in \Delta _{\text {sm}}\) that appears in (3.5).

### Lemma 3.2

The volume of the fiber over \(A\in \Delta _{\text {sm}}\) equals \(\vert p^{-1}(A)\vert = 2^{n}\pi \,(n-2)! .\)

### Proof

Let \(A= p(C,\mu ) \in \Delta _\mathrm {sm}\). The last coordinate \(\mu _{n-1}\) is always mapped to the double eigenvalue \(\lambda _{n-1}=\lambda _n\) of *A*, whereas there are \((n-2)!\) possibilities to arrange \(\mu _1,\ldots ,\mu _{n-2}\). For a fixed choice of \(\mu \) there are \(|O(1)|^{n-2}|O(2)|\) ways to choose \(C\in O(n)\). Therefore, by (3.3) we obtain \( \vert p^{-1}(A)\vert = \vert O(1)\vert ^{n-2} \vert O(2)\vert \,(n-2)!=2^{n-2}\cdot 2^2\pi \cdot (n-2)! = 2^{n} \pi (n-2)!\). \(\square \)

*u*. Considering the eigenvalues \(\mu _1,\ldots ,\mu _{n-2}\) as the eigenvalues of a symmetric \((n-2)\times (n-2)\) matrix

*Q*, by (3.2) we have

### Remark 6

### 3.3 Multiplicities in a Random Family

In this subsection we prove Theorem 1.2.

*a*and

*b*respectively and \(a+b\ge N-1\), then

## 4 The Second Moment of the Characteristic Polynomial of a GOE Matrix

In this section we give a proof of Theorem 1.6. Let us first recall some ingredients and prove some auxiliary results.

### Lemma 4.1

Let \(P_m=2^{1-m^2}\sqrt{\pi }^{m}\prod _{i=0}^m (2i)!\) and let \(Z_{2m}\) be the normalization constant from (3.1). Then \(P_m=2^{1-2m}\,Z_{2m}.\)

### Proof

*(physicist’s) Hermite polynomials*\(H_i(x),\, i=0,1,2, \ldots \) form a family of orthogonal polynomials on the real line with respect to the measure \(e^{-x^2}dx\). They are defined by

The following proposition is crucial for the proof of Theorem 1.6.

### Proposition 4.2

*k*and a fixed \(u\in \mathbb {R}\) the following holds.

- 1.If \(k=2m\) is even, thenwhere$$\begin{aligned} \mathop {\mathbb {E}}\limits \limits _{Q\sim \mathrm {GOE}(k)}\det (Q-u{\mathbb {1}})^2 = \frac{(2m)!}{2^{2m}}\,\sum _{j=0}^m \frac{2^{-2j-1}}{(2j)!}\, \det X_j(u), \end{aligned}$$$$\begin{aligned} X_j(u)=\begin{pmatrix} H_{2j}(u) &{} H_{2j}'(u) \\ H_{2j+1}(u)-H_{2j}'(u) &{} H_{2j+1}'(u)-H_{2j}''(u) \end{pmatrix}. \end{aligned}$$
- 2.If \(k=2m+1\) is odd, thenwhere$$\begin{aligned} \mathop {\mathbb {E}}\limits \limits _{Q\sim \mathrm {GOE}(k)}\det (Q-u{\mathbb {1}})^2 =\frac{\sqrt{\pi }(2m+1)!}{2^{4m+2}\,\Gamma (m+\tfrac{3}{2})} \sum _{j=0}^m \frac{2^{-2j-2}}{(2j)!}\, \det Y_j(u), \end{aligned}$$$$\begin{aligned} Y_j(u)=\begin{pmatrix} \frac{(2j)!}{j!} &{} H_{2j}(u) &{} H_{2j}'(u) \\ 0 &{} H_{2j+1}(u)-H_{2j}'(u) &{} H_{2j+1}'(u)-H_{2j}''(u)\\ \tfrac{(2m+2)!}{(m+1)!} &{} H_{2m+2}(u) &{} H_{2m+2}'(u) \end{pmatrix}. \end{aligned}$$

### Proof

In Section 22 of Mehta (2004) one finds two different formulas for the even \(k=2m\) and odd \(k=2m+1\) cases. We evaluate both separately.

Everything is now ready for the proof of Theorem 1.6

### Proof of Theorem 1.6

Due to the nature of Proposition 4.2 we have to make a case distinction also for this proof.

## Notes

### Acknowledgements

Open access funding provided by Max Planck Society. The authors wish to thank A. Agrachev, P. Bürgisser, A. Maiorana for helpful suggestions and remarks on the paper and B. Sturmfels for pointing out reference (Sanyal et al. 2013) for (1.2).

### Open Access

This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

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