Nonvanishing minors of eigenvector matrices and consequences

Tarek Emmrich

doi:10.1515/spma-2024-0020

Artikel Open Access

Nonvanishing minors of eigenvector matrices and consequences

Tarek Emmrich

Veröffentlicht/Copyright: 16. August 2024

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Abstract

For a matrix M ∈ K n × n , we establish a condition on the Galois group of the characteristic polynomial φ M that induces nonvanishing of the minors of the eigenvector matrix of M . For integer matrices, recent results by Eberhard show that, conditionally on the extended Riemann hypothesis, this condition is satisfied with high probability (We say “with high probability” for probability 1 − o ( 1 ) as n → ∞ .) and hence, with high probability, the minors of eigenvector matrices of random integer matrices are nonzero. For random graphs, this yields a novel uncertainty principle, related to Chebotarëv’s theorem on the roots of unity and results from Tao and Meshulam. We also show the application in graph signal processing and the connection to the rank of the walk matrix.

Keywords: nonvanishing minors; spectral graph theory

MSC 2010: 15-02

1 Introduction

For the Fourier matrix, there is Chebotarëv’s famous theorem on the nonvanishing of the minors, see the study by Stevenhagen and Lenstra [17] for a survey.

Theorem 1

[17] Let p be a prime and F be the Fourier matrix of order p , i.e., F i , j = ω ( i − 1 ) ( j − 1 ) for ω = e 2 π i p . Then all minors of F are nonzero.

From the viewpoint of spectral graph theory, the matrix F is a possible eigenvector matrix of the circle graph, and this yields the uncertainty principle

‖ f ‖ 0 + ‖ f ˆ ‖ 0 ≥ p + 1

for any function f on the circle graph and its Graph Fourier transform f ˆ = F * f , see Tao [18] and Meshulam [12] for uncertainty principles. For a field K of characteristic zero, we will prove the following criterion on nonvanishing of the minors of the eigenvector matrix of any matrix M ∈ K n × n . Let φ M be the characteristic polynomial of M and L its splitting field. We denote by Gal ( φ M ) the Galois group of the field extension K ⊆ L . Furthermore, if M is diagonalizable, let U be the matrix with columns u i that are eigenvectors corresponding to the eigenvalue λ i of M .

Theorem 2

Let U be the eigenvector matrix of a diagonalizable matrix M . If Gal ( φ M ) ≥ A n , then all minors of U are nonzero.

Note that for Gal ( φ M ) ≥ A n , M is always diagonalizable. Note further that this result only depends on the Galois group of the characteristic polynomial of M and hence also implies nonvanishing of all minors for BU for all B ∈ Gl n ( K ) . For various reasons, we care about the minors of the eigenvector matrices of operators on graphs, e.g., it induces an uncertainty principle for graph signals, it guarantees the success of different algorithms and has a connection to the walk matrix of a graph, see Chapters 3 and 4. The two widely studied operators on graphs are the adjacency matrix and the Laplacian matrix. For directed graphs, the directed adjacency matrix seems to be the most interesting.

For the probabilistic results, K = Q is an appropriate choice. For a fixed finitely supported measure μ on Z , let M be a random matrix with m i , j ∼ μ . In this case, Eberhard [5] proved that, conditionally on the extended Riemann hypothesis, with high probability, φ M is irreducible over Q and Gal ( φ M ) ≥ A n . This directly leads to the following corollary, which induces an uncertainty principle for directed graphs and their adjacency matrices.

Corollary 3

Assume the extended Riemann hypothesis. For a random matrix M , with m i , j ∼ μ , with high probability, all minors of the eigenvector matrix U of M are nonzero.

Following Eberhard’s results, Ferber et al. [7] could show that the Galois group of a random symmetric matrix is almost surely transitive, which we will see is not sufficient for minors of arbitrary size to be nonzero, but for minors of size one.

Corollary 4

Let M be the adjacency matrix of a random Erdős-Rényi graph G ∼ G ( n , p ) and assume the extended Riemann hypothesis. Then with high probability no eigenvector of M contains a zero.

Working with the Laplacian matrix of a random graph on n vertices has the difficulty that the rank of the Laplacian matrix of a connected graph is n − 1 by default. We can overcome this problem by using the factorization φ L ( λ ) = λ ⋅ ξ ( λ ) , but still known results for random matrices do not apply. For applications, the most interesting case is the following uncertainty principle that follows directly from our results.

Theorem 5

Let G be a graph on n vertices and φ L ( λ ) = ł ⋅ ξ ( λ ) . If Gal ( ξ ) ≥ A n − 1 , then all minors of U are nonzero, i.e.,

‖ f ‖ 0 + ‖ f ˆ ‖ 0 ≥ n + 1

for any signal f ≠ 0 on the graph G. Here, the Graph Fourier transform f ˆ is given by f ˆ = U * f .

The eigenvectors of random matrices with continuous distribution have been widely studied in the past, for a survey see [14], but as far as we know the minors have not been studied in the discrete case. In the discrete setting, the most common application are graphs. Brooks and Lindenstrauss [2] proved a nonlocalization for the eigenvectors of the normalized Laplacian matrix of large regular graphs, i.e., small subsets of vertices only contain a small part of the mass of an eigenvector. Alt et al. [1] determined regions for the edge probability p = d N of an Erdős-Rényi graph, such that the eigenvectors are delocalized. Both statements are relatively far away from a statement about minors. A very well-studied object is the walk matrix W of a graph, and we will establish a connection between its rank and the minors of U . The construction is well known in other areas of mathematics as Krylov spaces [16].

Outline. After introducing notations and the necessary background in Section 2, we will establish our main results and a slight variation of it in Section 3. We discuss several applications in Section 4. We end by extending the results to Laplacian matrices and by giving examples.

2 Background

In this section, we will shortly introduce some notation and the background of Galois theory, for a more detailed summary, see [13]. For a polynomial φ ∈ Q [ λ ] of degree n , the fundamental theorem of algebra tells us that φ has exactly n roots λ i , i = 1 , … , n , over the complex numbers C . Instead of working over the complex numbers, we will work over the splitting field L = Q ( λ i : i ∈ [ n ] ) of φ , which is the smallest field extension of Q that contains all roots of φ . The dimension of the splitting field is bounded by

dim Q ( L ) ≤ n ! .

The splitting field can also be constructed inductively by the step Q ⊆ K 1 ≔ Q [ λ ] ∕ R ( λ ) that adds at least one root of φ , where R ( λ ) is an irreducible factor of φ . After at most n steps, all roots of φ are contained in the resulting field L . The structure of this adding process is encoded in the Galois group

Gal ( φ ) ≔ Aut Q ( L ) ⊆ Sym ( { λ 1 , … , λ n } ) ≅ S n .

For fields of positive characteristic, the Galois group of an irreducible polynomial is always the cyclic group and hence too small for our results too hold. Moreover, by restricting to characteristic zero, we ensure that the field extension is separable and the Galois group is well defined. In most cases, we will interpret the Galois group of any polynomial as a group acting on the set [ n ] by the given isomorphism. In the generic case, over the rational numbers, Hilbert’s irreducibility theorem tells us that

Gal ( φ ) ≅ S n .

Two algebraic numbers μ 1 , μ 2 are called conjugate, if they have the same minimal polynomial φ over Q . For two conjugate algebraic numbers μ 1 , μ 2 , there is a g ∈ Gal ( φ ) with g ( μ 1 ) = μ 2 . We call a group H ⊆ Sym ( { λ 1 , … , λ n } ) m-homogeneous, if for all S , S ′ ∈ [ n ] m , there exists a g ∈ H such that

g ( { λ i ∣ i ∈ S } ) = { λ i ∣ i ∈ S ′ } .

We say a group H ⊆ Sym ( { λ 1 , … , λ n } ) is m-transitive, if this condition also holds for any ordered sets. The prior fact tells us that the Galois group of an irreducible polynomial is 1-homogeneous (which in this case equals transitivity). For a matrix U and W , S ⊆ [ n ] , we write U W , S for the submatrix of U with rows induced by W and columns induced by S .

For a tuple B = ( b 1 , … , b k ) ∈ C k , we define the Vandermonde matrix

V ( B ) = ( b j i ) j = 1 , … , k i = 0 , … , k − 1 .

It is well known that this matrix is invertible if and only if all values in B are distinct.

For a finite set V and E ⊆ V 2 , we call G = ( V , E ) a graph, and we often assume V = [ n ] . The elements of V are called the vertices and their degree is denoted by deg ( v ) . We say two vertices v , w are connected, if { v , w } ∈ E and sometimes write v ∼ w . For two vertices v , w , the distance d ( v , w ) is the smallest integer r such that there exist vertices v = v 0 , v 1 , … , v r = w with { v k , v k + 1 } ∈ E . The adjacency matrix A ∈ { 0 , 1 } n × n of G is defined by

A v , w = 1 , if { v , w } ∈ E , 0 , otherwise .

The combinatorial Laplacian matrix L ∈ Z n × n is the matrix L = D − A , with D v , v = deg ( v ) . For a connected graph, the rank of the Laplacian matrix is n − 1 with the kernel vector 1 .

A matrix U ∈ K n × n is orthonormal, if

U U * = U * U = I n .

For an orthonormal matrix U ∈ K n × n and a vector f ∈ K n , we denote

f ˆ ≔ U * f .

If U is the eigenvector matrix of the Laplacian matrix, f ˆ is called the Graph Fourier transform, with the inverse transformation

f = U f ˆ .

If U is not orthonormal, but only invertible, we can still define the analysis coefficients f ˆ = U − 1 f of f . We say a vector f ∈ K n is s -sparse, if the size of its support equals s . We denote the size of the support of a vector f also by

‖ f ‖ 0 ≔ # { i ∈ [ n ] : f ( i ) ≠ 0 } .

3 Minors and Galois groups

Let M ∈ K n × n be a diagonalizable matrix, and let φ M be its characteristic polynomial with splitting field L . We denote the eigenvalues and eigenvectors by λ 1 , … , λ n ∈ L and u 1 , … , u n ∈ L n . The eigenvector matrix U is the matrix with columns u i . For any two conjugate elements λ i , λ j , there exists g ∈ Gal ( φ M ) such that g ( λ i ) = λ j . By applying this g to the eigenvector equation M u i = λ i u i , we obtain

(1) g ( M ) g ( u i ) = λ j g ( u i ) .

Since M ∈ K n × n and g ∣ K is the identity, we have g ( M ) = M and hence the equation (1) tells us that g ( u i ) is a scalar multiple of u j . We denote by u i J the restriction of the ith eigenvector to the rows with indices in J . We are now ready to prove Theorem 2.

Proof of Theorem 2

Assume there are sets S , W ∈ [ n ] m for some m ∈ [ n ] such that det ( U W , S ) = 0 . Let a 1 , … , a m ∈ L be the coefficients such that

∑ i ∈ S a i u i W = 0 W .

Let S ′ ∈ [ n ] m , then there exists g ∈ Gal ( φ M ) with g ( { λ i ∣ i ∈ S } ) = { λ i ∣ i ∈ S ′ } . Applying g , we obtain

∑ i ∈ S ′ g ( a i ) g ( u i ) W = 0 W .

By equation (1), the vectors { g ( u i ) W : i ∈ S } are scalar multiples of the vectors { u i W : i ∈ S ′ } , and hence, the vectors { u i W : i ∈ S ′ } are linearly dependent. Since this holds for any S ′ ∈ [ n ] m , this yields rank ( U W , [ n ] ) = m − 1 , contradicting the linear independence of the rows of U .□

Eberhard [5, Theorem 1.3] proved that, conditionally on the extended Riemann hypothesis, the Galois group of the characteristic polynomial of a random matrix is almost surely A n or S n and hence Corollary 3 follows. Under the same condition Ferber et al. [7, Theorem 1.1] could show that the characteristic polynomial of a random symmetric matrix is almost surely irreducible. Thus, Corollary 4 follows.

Our condition Gal ( φ M ) ≥ A n is slightly stronger than just the m -homogeneity. Kantor [10] [Theorem 1] was able to show that only for m = 2 , 3 , 4 , there are groups that are m -homogeneous, but not m -transitive. In addition, by the classification of finite simple groups, it was proven that for m ≥ 6 , there is no m -transitive group other than A n or S n [3]. The most precise way to state the connection is the following.

Theorem 6

Let U be the eigenvector matrix of a diagonalizable matrix M with characteristic polynomial φ M . If the Galois group Gal ( φ M ) is m-homogeneous, then all minors of size m and ( n − m ) do not vanish. In particular, if the group Gal ( φ M ) is transitive, i.e., the polynomial φ M is irreducible, the matrix U does not contain a zero.

The proof is similar to the proof of Theorem 2. Notice that m -homogeneity is equivalent to ( n − m ) -homogeneity. The characteristic polynomial of the adjacency matrix of the graph in the following example is not 4-homogeneous.

Example 7

The characteristic polynomial of the adjacency matrix of the graph in Figure 1 has the characteristic polynomial

φ A = λ 8 − 10 λ 6 − 4 λ 5 + 23 λ 4 + 12 λ 3 − 12 λ 2 − 4 λ + 1 ,

which is irreducible, but with the Galois group

Gal ( φ A ) ≅ { σ ∈ S 8 : σ ( { 1 , 2 , 3 , 4 } ) ∈ { { 1 , 2 , 3 , 4 } , { 5 , 6 , 7 , 8 } } } .

By checking the minors of the eigenvector matrix, we see that several minors of size 4 are vanishing. These vanishing minors are supported on eigenvectors { u i : i ∈ S } with S = { 1 , 2 , 3 , 4 } or S = { 5 , 6 , 7 , 8 } .

$Figure 1 A graph with 1-homogeneous Galois group Gal ( φ A ) < A 8 {\rm{Gal}}\left({\varphi }_{{\bf{A}}})\lt {A}_{8} , but vanishing minors.$

Figure 1

A graph with 1-homogeneous Galois group Gal ( φ A ) < A 8 , but vanishing minors.

Remark 8

It is quite natural to ask if there is any connection between Chebotarëvs theorem and Theorem 2. The Fourier matrix can be seen as the eigenvector matrix of the shift matrix. Its characteristic polynomial is

λ p − 1 = ( λ − 1 ) ⋅ ∑ k = 0 p − 1 λ k ≔ λ ⋅ ξ ( λ )

with splitting field L = Q [ λ ] ∕ ξ ( λ ) and Galois group ( Z ∕ p Z ) × , which is not sufficient for our results to hold. This has the simple reason that for Chebotarëvs theorem on the Fourier matrix the explicit knowledge of the eigenvectors is used in the proof. There are matrices with characteristic polynomial ( λ p − 1 ) and also ( x p − 1 + … + x + 1 ) whose eigenvector matrices have vanishing minors, for example, for p = 3 let F be the Fourier matrix of order 3 and ω be a root of λ 2 + λ + 1 . By an easy change of basis, we obtain

B S 3 B − 1 = 1 1 1 0 1 0 0 0 1 0 0 1 1 0 0 0 1 0 1 − 1 − 1 0 1 0 0 0 1 = 1 0 0 1 − 1 − 1 0 1 0 ,

with the eigenvector matrix

U = BF = 3 0 0 1 ω ω 2 1 ω 2 ω .

Similarly for p > 3 , eigenvector matrices with bigger vanishing minors can be constructed.

The Galois group can not only be used for minors but also for slightly more structured problems. We will introduce a matrix G ( S , W , r 1 , … , r k ) and one might think of it as a variation of U W , S with # W < # S , but there are additional rows of higher order added. The problem of vanishing minors originally appeared in the analysis of an algorithm [6], which recovers sparse signals on graphs, whose success relies on the regularity of the matrix

G v , j = f v ( j )

with f v ( j ) = ∑ i ∈ S λ i j u i ( v ) for 0 ≤ j ≤ s − 1 and v ∈ W ⊆ [ n ] . The factorization G = U W , S ⋅ V ( λ i : i ∈ S ) yields the connection to minors. A slightly improved version of this algorithm relies on the regularity of the following matrix:

G ˜ = f 1 ( 0 ) f 1 ( 1 ) … f 1 ( s − 1 ) f 1 ( 1 ) f 1 ( 2 ) … f 1 ( s ) ⋮ ⋮ ⋮ f 1 ( r 1 − 1 ) f 1 ( r 1 ) … f 1 ( s − 2 + r 1 ) f 2 ( 0 ) f 2 ( 1 ) … f 2 ( s − 1 ) f 2 ( 1 ) f 2 ( 2 ) … f 2 ( s ) ⋮ ⋮ ⋮ f k ( r k − 1 ) f k ( r k ) … f k ( s − 2 + r k )

for some radii ∑ i = 1 k r i = m and corresponding vertices W = { v 1 , … , v k } . This matrix has the factorization

G ˜ = G ( S , W , r 1 , … , r k ) ⋅ V ( λ i : i ∈ S ) = u j 1 ( v 1 ) u j 2 ( v 1 ) … u j s ( v 1 ) λ j 1 u j 1 ( v 1 ) λ j 2 ( v 1 ) … l j s u j s ( v 1 ) ⋮ ⋮ ⋮ λ j 1 r 1 − 1 u j 1 ( v 2 ) λ j 2 r 1 − 1 u j 2 ( v 2 ) … λ j s r 1 − 1 u j s ( v 2 ) u j 1 ( v 2 ) u j 2 ( v 2 ) … u j s ( v 2 ) λ j 1 u j 1 ( v 2 ) λ j 2 ( v 2 ) … l j s u j s ( v 2 ) ⋮ ⋮ ⋮ λ j 1 r k − 1 u j 1 ( v k ) λ j 2 r k − 1 u j 2 ( v k ) … λ j s r k − 1 u j s ( v k ) ⋅ V ( λ i : i ∈ S ) ,

for W = { v i : 1 ≤ i ≤ k } and S = { j i : 1 ≤ i ≤ s } . Note that the rows of each vertex-block are linearly independent as soon as u j ( v ) ≠ 0 for j ∈ S . The next matrix is a small example of such a matrix:

G ( { 1 , 2 } , { v , w } , 2 , 2 ) = u 1 ( v ) u 2 ( v ) u 3 ( v ) u 4 ( v ) λ 1 u 1 ( v ) λ 2 u 2 ( v ) λ 3 u 3 ( v ) λ 4 u 4 ( v ) u 1 ( w ) u 2 ( w ) u 3 ( w ) u 4 ( w ) λ 1 u 1 ( w ) λ 2 u 2 ( w ) λ 3 u 3 ( w ) λ 4 u 4 ( w ) .

Theorem 9

Let U be the eigenvector matrix of A . The matrix G ( S , W , r 1 , … , r k ) has full rank, if Gal ( φ A ) ≥ A n and d ( v i , v j ) > r i + r j for all 1 ≤ i , j ≤ k .

Proof

Assume, by contradiction, that rank ( G ( S , W , r 1 , … , r k ) ) < s . Then there are coefficients a i such that

∑ i ∈ S a i λ i k i u i ( v ) = 0

for v ∈ W and 0 ≤ k i ≤ r i − 1 . Applying each γ ∈ Gal ( φ A ) yields rank ( G ( [ n ] , W , r 1 , … , r k ) ) < s . Let Z v , m be the row of G ( [ n ] , W , r 1 , … , r k ) belonging to vertex v and power m . We have

⟨ Z v , m , Z w , l ⟩ = ∑ i = 1 n λ i m + l u i ( v ) u i ( w ) = A v , w m + l .

And hence we know that ⟨ Z v , m , Z w , l ⟩ = 0 for d ( v , w ) > m + l . This yields that the rows of distinct blocks are orthogonal and hence implies that the intersection of the span of the blocks is 0, if the distance condition is fulfilled. Again, by an Vandermonde decomposition of the blocks, we see that the blocks have full rank and hence the matrix G ( [ n ] , W , r 1 , … , r k ) cannot be singular, a contradiction.□

4 Consequences of nonvanishing minors and applications

There are several consequences of the minors of the eigenvector matrix being nonzero. The first one is an uncertainty principle similar to the one from Tao.

Theorem 10

[18, Theorem 1.1] Let U ∈ K n × n be a matrix. All minors of any size of U are nonzero if and only if

‖ f ‖ 0 + ‖ f ˆ ‖ 0 ≥ n + 1 ,

for all 0 ≠ f ∈ K n .

Proof

We proof the if-part of the statement first by contradiction. Assume, some minor of U is zero. Then there are coefficients a i ∈ K such that ∑ i ∈ I a i u i J = 0 J for # I = # J = m . The vector

f ˆ ( i ) = a i for i ∈ I , 0 else ,

has support of size m . This choice yields ( U f ˆ ) J = 0 J , and hence,

‖ f ˆ ‖ 0 + ‖ U f ˆ ‖ 0 ≤ m + ( n − m ) = n .

Now again, by contradiction, assume ‖ f ‖ 0 + ‖ f ˆ ‖ 0 = n and denote ‖ f ‖ 0 = r . Let W = supp ( f ) be the support of f and S = supp ( f ˆ ) be the support of f ˆ with # S = n − r . Restricting U to the columns belonging to S yields

U [ n ] , S f ˆ S = f ,

which means that the minor U [ n ] \ W , S is zero, a contradiction.□

Another well-known consequence is the following application to sparse recovery.

Theorem 11

[8, Theorem 2.13] Let G be a graph with vertex set [ n ] and M = U Λ U * for M ∈ { L , A } with nonvanishing minors of U of any size. For any 1 ≤ s ≤ n , pick any W , S ∈ [ n ] s , then each s-sparse signal

f = ∑ w ∈ W a w δ w

is uniquely determined by at least 2s samples of the form

⟨ u i , f ⟩ .

Let U be the eigenvector matrix of M ∈ { L , A } and let 0 ≠ x ∈ R n be a vector. One interesting object to study is the so called walk matrix, see, e.g., [4,9,11,15],

W ( x ) ≔ ∣ ∣ ∣ ∣ ∣ x M x M 2 x ⋯ M n − 1 x ∣ ∣ ∣ ∣ ∣

and its rank, depending on x . Let 0 ≤ r ≤ n − 1 be the minimal number such that there are coefficients a 0 , … , a r with

∑ k = 0 r a k M k x = 0 [ n ] .

Then r is the degree of the polynomial Q ( λ ) = ∑ k = 0 r a k λ k and also the rank of W , because each set of columns with indices in { i , … , i + r } will have the kernel

( a 0 , … , a r ) T .

The next theorem describes the connection between the choice of x , the rank of the walk matrix, and the nonvanishing of minors of U .

Theorem 12

Let G be a graph on n vertices, 0 ≠ x ∈ R n , and let W ( x ) be the walk matrix for M = A .

If φ A is irreducible and x ∈ Q n , then
rank ( W ( x ) ) = n .
For more general 0 ≠ x ∈ R n , we have
rank ( W ( x ) ) = ‖ U * x ‖ 0 = ‖ x ˆ ‖ 0 .
If all minors of U are nonvanishing, the following formula holds
rank ( W ( x ) ) ≥ n + 1 − ‖ x ‖ 0 .

Proof

In a first step, we will show the following equation:

rank ( W ( x ) ) = n − ‖ U * x ‖ 0 .

Let a k be the entries of the kernel of W ( x ) and let Q = ∑ k = 0 r a k λ k be the corresponding polynomial, i.e.,

Q ( A ) ⋅ x = ∑ k = 0 r a k A k x = 0 [ n ] ,

which is equivalent to

(2) ∑ k = 0 r a k ∑ w = 1 n A v , w k x w = 0 for all v .

The diagonalization of A yields

A v , w k = ∑ l = 1 n u l ( v ) u l ( w ) λ l k ,

and we want to use this equation to obtain a formula for ( Q ( A ) ⋅ x ) that gives more insight

∑ k = 0 r a k ∑ w = 1 n A v , w k x w = ∑ k = 0 r a k ∑ w = 1 n ∑ l = 1 n u l ( v ) u l ( w ) λ l k x w = ∑ l = 1 n u l ( v ) ∑ w = 1 n x w u l ( w ) ∑ k = 0 r a k λ l k = ∑ l = 1 n u l ( v ) ∑ w = 1 n x w u l ( w ) Q ( λ l ) .

We can see now that equation (2) is equivalent to

∑ l = 1 n u l ( v ) Q ( λ l ) ∑ w = 1 n x w u l ( w ) = 0 for all v .

This condition reads as follows:

( u l ( v ) ) l = 1 , … , n , Q ( λ l ) ∑ w = 1 n x w u l ( w ) l = 1 , … , n = 0 for all v

and since the vectors ( u l ( v ) ) l = 1 , … , n form a basis this yields

Q ( λ l ) ∑ w = 1 n x w u l ( w ) = 0

for all l . The second vector can be written as follows:

∑ w = 1 n w x u l ( w ) l = 1 , … , n = U * x .

The polynomial Q has at most r distinct zeros, where r is also the rank of W ( x ) , and thus,

n = rank ( W ( x ) ) + ‖ U * x ‖ 0

holds, which is equivalent to the stated equation.

Now for the first point, observe that x ∈ Q n implies a k ∈ Q , and hence, a k ∈ Z , and thus, Q divides φ A in the polynomial ring Q [ X ] . If φ A is irreducible, either Q = φ A or Q = 1 . The case Q = 1 is not possible for x ≠ 0 and hence rank ( W ) = n for φ A irreducible and x ∈ Q n .

For not necessarily rational x ∈ R n , we see that

rank ( W ( x ) ) = ‖ U * x ‖ 0 = ‖ x ˆ ‖ 0 .

For the last point, since U has nonvanishing minors, Theorem 10 implies

rank ( W ( x ) ) ≥ n + 1 − ‖ x ‖ 0 .□

5 Laplacian matrices

We now formulate the analogous versions of the prior theorems for the Laplacian matrices with necessary changes. Let ξ ( λ ) be the unique polynomial with φ L = λ ⋅ ξ ( λ ) . We need a symmetry argument for matrices with orthogonal columns.

Lemma 13

Let U be the a matrix with orthogonal columns u i . If there are sets W , S ∈ [ n ] m such that the vectors ( u i W ) i ∈ S are linearly dependent, then the vectors ( u i [ n ] \ W ) i ∈ [ n ] \ S are also linearly dependent.

Proof

Let a i ∈ R some coefficients, such that

∑ i ∈ S a i u i W = 0 W .

For j ∉ S , we have ⟨ u j , u i ⟩ = 0 for all i ∈ S , and it follows that

0 = u j , ∑ i ∈ S a i u i = u j W , ∑ i ∈ S a i u i W + u j [ n ] \ W , ∑ i ∈ S a i u i [ n ] \ W = 0 + u j [ n ] \ W , ∑ i ∈ S a i u i [ n ] \ W .

Since the vectors u i are linearly independent we know that ∑ i ∈ S a i u i ≠ 0 [ n ] , and hence,

∑ i ∈ S a i u i [ n ] \ W ≠ 0 [ n ] \ W .

Now the ( n − m ) vectors u j [ n ] \ W of length ( n − m ) lie in the hyperplane orthogonal to ∑ i ∈ S a i u i [ n ] \ W and therefore are linearly dependent.□

This lemma is all we need to prove the following theorem for Laplacian matrices.

Theorem 14

Let G be a graph on n vertices and U be the eigenvector matrix of the Laplacian matrix L with characteristic polynomial φ L = λ ⋅ ξ . If Gal ( ξ ) ≥ A n − 1 , then we have

det ( U W , S ) ≠ 0

for all W , S ∈ [ n ] m .

Proof

By Lemma 13, we can assume 1 ∉ S . If det ( U W , S ) = 0 for some W , S ∈ [ n ] m , we know that det ( U W , S ′ ) = 0 for all S ′ ∈ { 2 , … , n } m by applying the m -homogeneous Galois group to the equation, and hence,

rank ( U W , { 2 , … , n } ) = m − 1 .

Thus, there are coefficients a w ∈ K for w ∈ W such that

∑ w ∈ W a w u ˜ ( w ) = 0 ,

where u ˜ ( w ) = ( u i ( w ) ) i = 2 , … , n . For v ∉ W , we know that ⟨ u ( v ) , u ( w ) ⟩ = 0 , and hence,

0 = u ( v ) , ∑ w ∈ W a w u ( w ) = u 1 ( v ) , ∑ w ∈ W a w u 1 ( w ) + u ˜ ( v ) , ∑ w ∈ W a w u ˜ ( v ) = u 1 ( v ) ⋅ ∑ w ∈ W a w u 1 ( w ) .

Now since the vectors u ( w ) are linearly independent for w ∈ W , we know that ∑ w ∈ W a w u 1 ( w ) ≠ 0 and together with u 1 ( v ) = 1 n ≠ 0 this yields a contradiction.□

Theorem 5 follows now from Theorem 14 and the fact that A n − 1 and S n − 1 are m -homogeneous, while the equivalence of nonvanishing minors and uncertainty principle has been shown in Theorem 10. Moreover, if ξ is irreducible, it also follows directly from Theorem 14 that all eigenvectors do not contain a zero.

Theorem 15

Let G be a graph on n vertices, 0 ≠ x ∈ R n , and let W ( x ) be the walk matrix for M = L with characteristic polynomial φ L = λ ⋅ ξ .

If ξ ( λ ) is irreducible and x ∈ Q n , then
rank ( W ( x ) ) = n .
For more general 0 ≠ x ∈ R n , we have
rank ( W ) = ‖ U * x ‖ 0 = ‖ x ‖ 0 .
If all minors of U are nonvanishing, the following formula holds
rank ( W ) ≥ n + 1 − ‖ x ‖ 0 .

The only change in the proof in comparison to Theorem 12 is the use of λ 1 = 0 and u 1 = 1 . The following theorem is the analogous version of Theorem 9 for Laplacian matrices.

Theorem 16

Let G be a graph on n vertices and U be the eigenvector matrix of the Laplacian matrix L with characteristic polynomial φ = λ ⋅ ξ . The matrix G ( S , W , r 1 , … , r k ) has full rank, if Gal ( ξ ) ≥ A n − 1 and d ( v i , v j ) > r i + r j for all 1 ≤ i , j ≤ k .

Proof

By a similar argument as in the proof of Theorem 9, we obtain that singularity of G ( S , W , r 1 , … , r k ) yields singularity of G ( { 2 , … , n } , W , r 1 , … , r k ) . Let Z v , m be the row of G ( { 2 , … , n } , W , r 1 , … , r k ) belonging to vertex v and power m . For m + l > 0 , we have

⟨ Z v , m , Z w , l ⟩ = ∑ i = 2 n λ m + l u i ( v ) u i ( w ) = L v , w m + l .

The radius condition d ( v i , v j ) > r i + r j yields orthogonality for m + l > 0 . The rows belonging to power 0 cannot be linearly dependent according to Theorem 14.□

We finally provide two examples.

Example 17

The Laplacian matrix of the graph in Figure 2 has the characteristic polynomial

φ = λ ⋅ ( λ 8 − 28 λ 7 + 332 λ 6 − 2170 λ 5 + 8516 λ 4 − 20440 λ 3 + 29105 λ 2 − 22288 λ + 6957 ) .

The Galois group of ξ has order 384 and is given by

Gal ( ξ ) ≅ σ ∈ S 8 : σ ( i + 4 ) = σ ( i ) + 4 , for σ ( 1 ) ≤ 4 , σ ( i ) − 4 , for σ ( 1 ) ≥ 5 , for 1 ≤ i ≤ 4 .

This group is imprimitive but the minors of any size are nonzero. We see that our criterion is not necessary.

$Figure 2 A graph with imprimitive Galois group Gal ( ξ ) < A 8 {\rm{Gal}}\left(\xi )\lt {A}_{8} , but nonvanishing minors.$

Figure 2

A graph with imprimitive Galois group Gal ( ξ ) < A 8 , but nonvanishing minors.

Example 18

The Laplacian matrix of the graph in Figure 3 has the characteristic polynomial

φ = λ ⋅ ( λ 6 − 12 λ 5 + 54 λ 4 − 114 λ 3 + 115 λ 2 − 50 λ + 7 ) .

The Galois group of ξ has order 72 and is given by

Gal ( ξ ) ≅ { σ ∈ S 6 : σ ( { 1 , 2 , 3 } ) ∈ { { 1 , 2 , 3 } , { 4 , 5 , 6 } } } .

This group is imprimitive and the vanishing minors of size 3 are supported on the columns { 1 , 2 , 3 } or { 4 , 5 , 6 } . This means that the condition on the Galois group cannot be weakened.

Figure 3

A graph with 1-homogeneous Galois group, but vanishing minors.

Acknowledgement

I thank the anonymous reviewers for their helpful comments and suggestions that really helped to improve some parts of the manuscript. Furthermore, I thank Martina Juhnke and Stefan Kunis for many fruitful discussions and proofreading early versions of this manuscript.

Funding information: This research received no specific grant from any funding agency.
Author contribution: The author confirms the sole responsibility for the conception of the study, presented results and manuscript preparation.
Conflict of interest: The author declares no conflict on interest.
Data availability statement: Data sharing is not applicable to this article as no datasets were generated or analysed during the current study.

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Received: 2024-03-20

Revised: 2024-06-14

Accepted: 2024-07-08

Published Online: 2024-08-16

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https://doi.org/10.1515/spma-2024-0020

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nonvanishing minors; spectral graph theory

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