I'm trying to get insight into the eigenvalue spectrum of a square matrix (large N, symmetric, positive semi-definite matrix) using Fourier transforms (I've tried transforming a bunch of things: the diagonal, the skew diagonal, the entire matrix, each eigenvector, etc. . Is there anything I should be looking at specifically?
2026-05-05 22:11:16.1778019076
Is it possible to link the eigenvalues of a matrix to the Fourier transform of the matrix?
5.1k Views Asked by user7815 https://math.techqa.club/user/user7815/detail At
1
There are 1 best solutions below
Related Questions in MATRICES
- How to prove the following equality with matrix norm?
- I don't understand this $\left(\left[T\right]^B_C\right)^{-1}=\left[T^{-1}\right]^C_B$
- Powers of a simple matrix and Catalan numbers
- Gradient of Cost Function To Find Matrix Factorization
- Particular commutator matrix is strictly lower triangular, or at least annihilates last base vector
- Inverse of a triangular-by-block $3 \times 3$ matrix
- Form square matrix out of a non square matrix to calculate determinant
- Extending a linear action to monomials of higher degree
- Eiegenspectrum on subtracting a diagonal matrix
- For a $G$ a finite subgroup of $\mathbb{GL}_2(\mathbb{R})$ of rank $3$, show that $f^2 = \textrm{Id}$ for all $f \in G$
Related Questions in FOURIER-SERIES
- order of zero of modular form from it's expansion at infinity
- Fourier series expansion of $\frac{\pi^4}{96}$ and $\frac{\pi^4}{90}$
- How is $\int_{-T_0/2}^{+T_0/2} \delta(t) \cos(n\omega_0 t)dt=1$ and $\int_{-T_0/2}^{+T_0/2} \delta(t) \sin(n\omega_0 t)=0$?
- Fourier series. Find the sum $\sum_{n=1}^\infty \frac{(-1)^{n+1}}{2n+1}$
- How get a good approximation of integrals involving the gamma function, exponentials and the fractional part?
- The convolution theorem for fourier series.:$ \widehat{f*g}(x) =2π\hat{g}(x)\cdot\hat{f}(x) $
- Ergodicity of a skew product
- Fourier Series on $L^1\left(\left[0,1\right)\right)\cap C\left(\left[0,1\right)\right)$
- Parseval's Identity Proof Monotone/Dominated Convergence Theorem
- How can I interchange the sum signs
Related Questions in EIGENVALUES-EIGENVECTORS
- Stability of system of parameters $\kappa, \lambda$ when there is a zero eigenvalue
- Stability of stationary point $O(0,0)$ when eigenvalues are zero
- Show that this matrix is positive definite
- Is $A$ satisfying ${A^2} = - I$ similar to $\left[ {\begin{smallmatrix} 0&I \\ { - I}&0 \end{smallmatrix}} \right]$?
- Determining a $4\times4$ matrix knowing $3$ of its $4$ eigenvectors and eigenvalues
- Question on designing a state observer for discrete time system
- Evaluating a cubic at a matrix only knowing only the eigenvalues
- Eigenvalues of $A=vv^T$
- A minimal eigenvalue inequality for Positive Definite Matrix
- Construct real matrix for given complex eigenvalues and given complex eigenvectors where algebraic multiplicity < geometric multiplicity
Trending Questions
- Induction on the number of equations
- How to convince a math teacher of this simple and obvious fact?
- Find $E[XY|Y+Z=1 ]$
- Refuting the Anti-Cantor Cranks
- What are imaginary numbers?
- Determine the adjoint of $\tilde Q(x)$ for $\tilde Q(x)u:=(Qu)(x)$ where $Q:U→L^2(Ω,ℝ^d$ is a Hilbert-Schmidt operator and $U$ is a Hilbert space
- Why does this innovative method of subtraction from a third grader always work?
- How do we know that the number $1$ is not equal to the number $-1$?
- What are the Implications of having VΩ as a model for a theory?
- Defining a Galois Field based on primitive element versus polynomial?
- Can't find the relationship between two columns of numbers. Please Help
- Is computer science a branch of mathematics?
- Is there a bijection of $\mathbb{R}^n$ with itself such that the forward map is connected but the inverse is not?
- Identification of a quadrilateral as a trapezoid, rectangle, or square
- Generator of inertia group in function field extension
Popular # Hahtags
second-order-logic
numerical-methods
puzzle
logic
probability
number-theory
winding-number
real-analysis
integration
calculus
complex-analysis
sequences-and-series
proof-writing
set-theory
functions
homotopy-theory
elementary-number-theory
ordinary-differential-equations
circles
derivatives
game-theory
definite-integrals
elementary-set-theory
limits
multivariable-calculus
geometry
algebraic-number-theory
proof-verification
partial-derivative
algebra-precalculus
Popular Questions
- What is the integral of 1/x?
- How many squares actually ARE in this picture? Is this a trick question with no right answer?
- Is a matrix multiplied with its transpose something special?
- What is the difference between independent and mutually exclusive events?
- Visually stunning math concepts which are easy to explain
- taylor series of $\ln(1+x)$?
- How to tell if a set of vectors spans a space?
- Calculus question taking derivative to find horizontal tangent line
- How to determine if a function is one-to-one?
- Determine if vectors are linearly independent
- What does it mean to have a determinant equal to zero?
- Is this Batman equation for real?
- How to find perpendicular vector to another vector?
- How to find mean and median from histogram
- How many sides does a circle have?
I know of a couple of results for certain special types of matrices.
1) If you have a circulant matrix, the eigenvalues of that matrix will be the discrete Fourier transform of the first row of the circulant matrix.
2) For the second one I don't remember all the technical details. Suppose you have a symmetric Toeplitz matrix whose size goes to infinity, certain functions of the eigenvalues can be written in terms of the Fourier transform of the first row of the matrix. In fact, the expression is written as a Riemann sum, which becomes a Riemann integral in the Fourier domain. In other words, you can write an unwieldy expression involving the eigenvalues of this growing matrix in terms of an integral which is usually much easier to evaluate.
If you want to learn more, check out Robert Gray's notes at http://ee.stanford.edu/~gray/toeplitz.pdf for an introduction to these kinds of results.
There is also the book by Grenander and Szego which gives a lot more detail and proves some more general results here
http://books.google.com/books?id=CFhVdL78wGcC&printsec=frontcover&dq=toeplitz+forms&hl=en&ei=CudxTf_XFtL1gAfH9qRV&sa=X&oi=book_result&ct=result&resnum=1&ved=0CDQQ6AEwAA#v=onepage&q&f=false