## How do I show that the eigenvalues of two square matrices of different dimensions are the same?

I have three matrices in a field $$F$$:

$$X in F^{a,a}, Y in F^{b,b}, Z in F^{a,b}$$, where $$a,b in mathbb{N}, a geq b$$ and $$text{rank}(Z) = b$$. The following term describes their relation:

$$A cdot C = C cdot B$$

I want to show that they have the same eigenvalues. I have started with the following about their determinant:

$$text{det}(AC) = text{det}(CB) iff text{det}(A) = text{det}(B)$$

However, I am not even sure if this is even relevant and how to go on from there.

## eigenvalues – How to find graph with specific order?

I worked on a problem in my research. I have a graph, $$G$$, with $$2n$$ vertices. It has one connected component of order $$2n-1$$ and an isolated vertex. $$lambda_1geq lambda_2geq ldots geq lambda_{2n}$$ are the eigenvalues of $$G$$. I have some bounds for them.
$$2n-3leq lambda_1<2n-2,\ 0leq lambda_2leq 1,\ -1leq lambda_ileq frac{1}{2}, 3leq i leq n+1,\ -3leq lambda_{n+2}leq -1,\ -3leq lambda_ileq frac{-3}{2}, n+ 3leq i leq 2n.$$
Also the maximum degree of this graph is $$2n-2$$.
How to find these graphs for $$6leq n leq 10$$ ?

## matrices – Eigenvalues of large size “identity” matrix

In the context of AR(1) model, the following $$n times n$$ matrix plays an important role:

$$V(rho) = {rho^{|i – j|}}_{1 leq i, j leq n}$$

(rho in (0, 1))
.
\$\$

I am interested in asymptotic properties of the following:

$$hat I_n := V(rho)^{1/2} V(hatrho)^{-1} V(rho)^{1/2}$$

where $$hat rho$$ is an estimator of $$rho.$$
Intuitively, this matrix is close to the $$n times n$$ identity matrix, but the problem is that the size $$n$$ grows, so we cannot simply write like $$hat I_n to I_n.$$
Still, I believe that $$hat I_n$$ is close to the identity matrix in a sense; for instance, all the eigenvalues go to one.

Let $$0 leq lambda^{(n)}_1 leq cdots leq lambda^{(n)}_n$$ be the eigenvalues of $$hat I_n.$$
My conjectures are

(1) for fixed $$i,$$ $$lambda^{(n)}_i overset{p}{to}1;$$

(2) $$lambda^{(n)}_n overset{p}{to} 1$$ and $$lambda^{(n)}_1 overset{p}{to} 1;$$

(3) (hopefully) $$sqrt{n} (lambda^{(n)}_n – 1) overset{d}{to} text{some distribution}$$ and $$sqrt{n} (lambda^{(n)}_1 – 1) overset{d}{to} text{some distribution}.$$

Are these correct under some conditions? Thanks!

## Closed geodesics and eigenvalues in a non-regular graph

Let $$Gamma$$ be a graph the degree of whose $$n$$ vertices is $$leq D$$ without necessarily being constant. Say we have bounds of type $$leq gamma^{2 k}$$ for the number of closed geodesics of length $$2 k$$ for any large $$k$$, for some $$gamma$$. Can we bound the non-trivial eigenvalues of the adjacency matrix $$A$$ of $$Gamma$$?

(If the degree were constant, this would be easy, via the Ihara zeta function and/or Hashimoto’s operator. When the degree is non-constant, the relation between the Ihara zeta function, on the one hand, and the eigenvalues of $$A$$, on the other, is less clean.)

If it helps, you can assume $$gamma$$ is of size $$O(sqrt{D})$$.

## Eigenvalues of a matrix product

Given two matrices $$A$$ and $$B$$, how to obtain the eigenvalues of a matrix given by its product $$C= AB$$?

## linear algebra – Eigenvalues of block matrix

Given a positive definite matrix $$A in mathbb{R}^{ntimes n}$$ and a general matrix $$B in mathbb{R}^{mtimes n}$$, can I say somehing about the eigenvalues of

$$T = begin{bmatrix} alpha A & alpha B^T \ beta B & 0 end{bmatrix}$$,

with $$alpha, beta in mathbb{R}$$? Can I maybe give bounds of the eigenvalues of $$T$$ as a function of $$alpha, beta$$?

## fa.functional analysis – Lower-bounding the eigenvalues of a certain positive-semidefinite kernel matrix, as a function of the norm of the input matrix

Let $$phi:(-1,1) to mathbb R$$ be a function such that

• $$phi$$ is $$mathcal C^infty$$ on $$(-1,1)$$.
• $$phi$$ is continuous at $$pm 1$$.

For concreteness, and if it helps, In my specific problem I have $$phi(t) := t cdot (pi – arccos(t)) + sqrt{1-t^2}$$.

Now, given a $$k times d$$ matrix $$U$$ with linearly independent rows, consider the $$k times k$$ positive-semidefinite matrix $$C_U=(c_{i,j})$$ defined by $$c_{i,j} := K_{phi}(u_i,u_j)$$, where

$$K_phi(x,y) := |x||y|phi(frac{x^top y}{|x||x|})$$

Question. How express the eigenvalues of $$C$$ in terms of $$U$$ and $$phi$$ ?

I’m ultimated interested in lower-bounding $$lambda_{min}(C_U)$$ in terms of some norm of $$U$$ (e.g spectral norm or Frobenius norm).

Let $$X$$ be the $$(d-1)$$-dimensional unit-sphere in $$mathbb R^d$$, equipped with its uniform measure $$sigma_{d-1}$$, and consider the integral operator $$T_phi: L^{2}(X) to L^2(X)$$ defined by
$$T_{phi}(f):x mapsto int K_{phi}(x,y)f(y)dsigma_{d-1}(y).$$
It is easy to see that $$T_phi$$ is a compact positive-definite operator.

Question. Are the eigenvalues of $$C_U$$ be expressed as a function of (eigenvalues of) $$K_{phi}$$ ?

## eigenvalues – How to make a cross-correlation between 2 Fisher matrices from a pure mathematical point of view?

Firstly, I want to give you a maximum of informations and precisions about my issue. If I can’t manage to get the expected results, I will launch a bounty, maybe some experts or symply people who have been already faced to a similar problem would be able to help me.

1)

I have 2 covariance matrices known $$Cov_1$$ and $$Cov_2$$ that I want to cross-correlate. (Covariance matrix is the inverse of Fisher matrix).

I describe my approach to cross-correlate the 2 covariance matrices (the constraints are expected to be better than the constraints infered from a “simple sum” (elements by elements) of the 2 Fisher matrices).

• For this, I have performed a diagonalisation of each Fisher matrix $$F_1$$ and $$F_2$$ associated of Covariance matrices $$Cov_1$$ and $$Cov_2$$.

• So, I have 2 different linear combinations of random variablethat are uncorraleted, i.e just related by eigen values ($$1/sigma_i^2$$) as respect of their combination.

These eigen values of diagonalising are contained into diagonal matrices $$D_1$$ and $$D_2$$.

2) I can’t build a “global” Fisher matrix directly by summing the 2 diagonal matrices since the linear combination of random variables is different between the 2 Fisher matrices.

I have eigen vectors represented by $$P_1$$ and $$P_2$$ matrices.

That’s why I think that I could perform a “global” combination of eigen vectors where I can respect the MLE (Maximum Likelihood Estimator) as each eigen value :

$$dfrac{1}{sigma_{hat{tau}}^{2}}=dfrac{1}{sigma_1^2}+dfrac{1}{sigma_2^2}quad(1)$$

because $$sigma_{hat{tau}}$$ corresponds to the best estimator from MLE method.

So, I thought a convenient linear combination of each eigen vectors $$P_1$$ and $$P_2$$ that could allow to achieve it would be under a new matrix P whose each column represents a new eigein global vector like this :

$$P = aP_1 + bP_2$$

3) PROBLEM: : But there too, I can’t sum eigen values under the form $$D_1 + D_2$$ since the new matrix $$P= a.P_1 + b.P_2$$ can’t have in the same time the eigen values $$D_1$$ and also $$D_2$$ eigen_values, can it ?

I mean, I wonder how to build this new diagonal matrix $$D’$$ such that I could write :

$$P^{-1} cdot F_{1} cdot P + P^{-1} cdot F_{2} cdot P=D’$$

If $$a$$ and $$b$$ could be scalars, I could for example to start from taking the relation :

$$P^{-1} cdot F_{1} cdot P = a^2*D_1quad(1)$$

and $$P^{-1} cdot F_{2} cdot P = b^2*D_2quad(2)$$

with $$(1)$$ and $$(2)$$ making appear the relation : $$Var(aX+bY) = a^2 Var(X) + b^2 Var(Y) + 2ab Cov(X,Y) = a^2 Var(X) + b^2 Var(Y)$$ since we are in a new basis $$P$$ that respect $$(1)$$ and $$(2)$$.

But the issue is that $$a$$ and $$b$$ seems to be matrices and not scalars, so I don’t know how to proceed to compute $$D’$$.

4) CONCLUSION :

Is this approach correct to build a new basis $$P = a.P_1 + b.P_2$$ and $$D’ = a.a.D_1 + b.b.D_2$$ assuming $$a$$ and $$b$$ are matrices ?

The key point is : if I can manage to build this new basis, I could return back to the starting space, the one of single parameters (no more combinations of them) by simply doing :

$$F_{text {cross}}=P . D’ cdot P^{-1}$$ and estimate the constraints with covariance matrix : $$C_{text{cross}}=F_{text {cross}}^{-1}$$.

If my approach seems to be correct, the most difficulty will be to determine $$a$$ and $$b$$ parameters (which is under matricial form, at least I think since with scalar form, there are too many equations compared to 2 unknown).

Sorry if there is no code for instant but I wanted to set correctly the problematic of this approach before trying to implement.

Hoping I have been clear enough.

Any help/suggestion/track/clue is welcome to solve this problem, this would be fine to tell it.

## matrix analysis – Growth of eigenvalues for certain sequences of matrices

Suppose we have an aperiodic matrix $$A_t$$ that has entries that are either $$0$$ or are positive integer powers of $$t$$, i.e. we could have
$$A_t = begin{pmatrix} 0 & t & t^2\ t & t^2 & 0\ t & 0 & t end{pmatrix}$$
for example.

Suppose $$t>0$$ and let $$Lambda(t)$$ denote the unique, real, simple maximal eigenvalue of $$A_t$$ guaranteed by the Perron-Frobenius Theorem. If we consider the function
$$f(t) = logLambda(e^t)$$
then it is possible to show using a variational principle and perturbation theory that $$f(t)$$ is increasing, convex and analytic (this is non-trivial!) with uniformly bounded (for $$tinmathbb{R}$$) first derivative. In particular the limits
$$lim_{ttoinfty} frac{f(t)}{t} = alpha_1 text{and} lim_{tto – infty} frac{f(t)}{t} = alpha_2$$
both exist and are finite. My question is the following:
can we calculate the error term associated to these limits? That is, can we find $$g(t)$$ such that
$$f(t) = alpha_1 t + O(g(t))$$
as $$ttoinfty$$ for example?

Any thoughts/insights would be greatly appreciated – thanks!

## eigenvalues – Sorting Eigensystem According to Complicated Rule

I have looked for an answer to this but the near duplicates I could find seemed slightly distinct.

I have a matrix $$A$$ which has eigenvalues in pairs $$lambda_1,-lambda_1,lambda_2,-lambda_2,dots$$. I would like to sort the eigensystem such that the eigenvectors are in this order, with the eigenvalues having descending real parts. That is, I want to sort in descending order of the function $$f=|Re(cdot)|$$ and break ties by $$g=Re(cdot)$$.

What I was hoping for was something like:

``````f(z_) := Abs(Re(z));
g(z_) := Re(z);
{eval,evec} = SortBy(Eigensystem(N(A))(Transpose),{f,z})(Transpose);

``````

but this doesn’t work. Replacing {f,g} with Abs@*Re does work but not for the tiebreak (neither does {Abs@*Re,Re}).