MHB Inverse Eigenvalues: A Puzzling Question?

mathmari
Gold Member
MHB
Messages
4,984
Reaction score
7
Hey! :o

Does it stand that the eigenvalues of $A^{-T}A^{-1}$ are the inverse of the eigenvalues of $A^TA$ ?? (Wondering)
 
Physics news on Phys.org
It is as followed, right??

The eigenvalues of $A^{-T}A^{-1}$ are the inverse of the eigenvalues of $(A^{-T}A^{-1})^{-1}=AA^{T}$ and the eigenvalues of $AA^{T}$ are the same as the eigenvalues of $A^{T}A$. So the eigenvalues of $A^{-T}A^{-1}$ are the inverse of the eigenvalues of $A^{T}A$.

(Wondering)

But for which matrices $A$ does this stand?? (Wondering)
 
mathmari said:
It is as followed, right??

The eigenvalues of $A^{-T}A^{-1}$ are the inverse of the eigenvalues of $(A^{-T}A^{-1})^{-1}=AA^{T}$ and the eigenvalues of $AA^{T}$ are the same as the eigenvalues of $A^{T}A$. So the eigenvalues of $A^{-T}A^{-1}$ are the inverse of the eigenvalues of $A^{T}A$.

(Wondering)

But for which matrices $A$ does this stand?? (Wondering)
As long as $A$ is invertible, the proposition is valid as you have shown.
 

Thread 'Trouble understanding an online solution to an exercise in Dummit & Foote'

##\textbf{Exercise 10}:## I came across the following solution online: Questions: 1. When the author states in "that ring (not sure if he is referring to ##R## or ##R/\mathfrak{p}##, but I am guessing the later) ##x_n x_{n+1}=0## for all odd $n$ and ##x_{n+1}## is invertible, so that ##x_n=0##" 2. How does ##x_nx_{n+1}=0## implies that ##x_{n+1}## is invertible and ##x_n=0##. I mean if the quotient ring ##R/\mathfrak{p}## is an integral domain, and ##x_{n+1}## is invertible then...
View full post »

Thread 'Questions about non existence of GCDs vs (coimages, cokernels)'

The following are taken from the two sources, 1) from this online page and the book An Introduction to Module Theory by: Ibrahim Assem, Flavio U. Coelho. In the Abelian Categories chapter in the module theory text on page 157, right after presenting IV.2.21 Definition, the authors states "Image and coimage may or may not exist, but if they do, then they are unique up to isomorphism (because so are kernels and cokernels). Also in the reference url page above, the authors present two...
View full post »

Thread 'Decomposition into irreps of compact Lie group'

When decomposing a representation ##\rho## of a finite group ##G## into irreducible representations, we can find the number of times the representation contains a particular irrep ##\rho_0## through the character inner product $$ \langle \chi, \chi_0\rangle = \frac{1}{|G|} \sum_{g\in G} \chi(g) \chi_0(g)^*$$ where ##\chi## and ##\chi_0## are the characters of ##\rho## and ##\rho_0##, respectively. Since all group elements in the same conjugacy class have the same characters, this may be...
View full post »

Similar threads

I Getting eigenvalues of an arbitrary matrix with programming
Replies
1
Views
2K
I Unravelling Structure of a Symmetric Matrix
Replies
5
Views
2K
I Can one find a matrix that's 'unique' to a collection of eigenvectors?
Replies
33
Views
669
I Find the Eigenvalues and eigenvectors of 3x3 matrix
Replies
12
Views
2K
I Eigenproblem for non-normal matrix
Replies
5
Views
3K
MHB Finding Matrix D Without Calculating P Inverse: Help Appreciated!
Replies
2
Views
2K
Prove 9 is eigenvalue of ##T^2\iff## 3 or -3 eigenvalue of ##T##.
Replies
5
Views
2K
I Question about an eigenvalue problem: range space
Replies
1
Views
2K
MHB When there is a double root for the eigenvalue, how many eigenvectors?
Replies
1
Views
2K
B Is the Null Space of an Operator Defined by Its Zero Eigenvalue?
Replies
4
Views
5K

Hot Threads

Recent Insights

Back
Top