Dtjytyk

How to copy the rest of lines of a file to another file

Smooth vector fields on a surface modulo diffeomorphisms

Is this Paypal Github SDK reference really a dangerous site?

I am the person who abides by rules, but breaks the rules. Who am I?

How can I portion out frozen cookie dough?

Can the Witch Sight warlock invocation see through the Mirror Image spell?

What should I do when a paper is published similar to my PhD thesis without citation?

Why is there an extra space when I type "ls" on the Desktop?

Is "cogitate" used appropriately in "I cogitate that success relies on hard work"?

Trocar background-image com delay via jQuery

How should I solve this integral with changing parameters?

Why do phishing e-mails use faked e-mail addresses instead of the real one?

ESPP--any reason not to go all in?

How to write a chaotic neutral protagonist and prevent my readers from thinking they are evil?

Either of .... (Plural/Singular)

What is better: yes / no radio, or simple checkbox?

Finding the minimum value of a function without using Calculus

What do you call someone who likes to pick fights?

How exactly does an Ethernet collision happen in the cable, since nodes use different circuits for Tx and Rx?

Is it appropriate to ask a former professor to order a book for me through an inter-library loan?

Graphic representation of a triangle using ArrayPlot

Under what conditions can the right to remain silent be revoked in the USA?

School performs periodic password audits. Is my password compromised?

Is there stress on two letters on the word стоят

Eigen-matrix of a tensor?

Eigen values of a certain type of block matrixFinding the eigen vectors of a 3x3 matrixGiven an eigen values evaluate $S*tinybegin{bmatrix} 0\1\0 end{bmatrix}$Same eigen values giving 2 different eigen vectorsFind the eigenvalues and corresponding eigen vectors of the matrixFinding the a Jordan Basis for a matrix with 2 eigen vectors and one generalized eigen vectorfinding eigen vector for double eigen valuefast way to find eigen vector and eigen value by inspectionProduct of matrix and a vector upper bounded by product of eigenvalue and vectorEigen values of this block matrix?

1

$begingroup$

Short vision of my question:

Is there such a thing as eigen-matrix of a tensor?

---

Long vision of my question:

I was given

begin{equation}
A=begin{bmatrix}
2b-a & a-b\
2b-2a & 2a-b\
end{bmatrix}
end{equation}

to find its eigenvalue and eigenvector, and I have found them with standard method.

However, I observed that the first column vector of $A$ is:

begin{equation}
begin{bmatrix}
2b-a\
2b-2a\
end{bmatrix} =begin{bmatrix}
-1& 2\
-2 & 2\
end{bmatrix}
begin{bmatrix}
a\
b
end{bmatrix}=Lambda_1s
end{equation}

similarly for the second column:

begin{equation}
begin{bmatrix}
a-b\
2a-b\
end{bmatrix} =begin{bmatrix}
1& -1\
2 & -1\
end{bmatrix}
begin{bmatrix}
a\
b
end{bmatrix}=Lambda_2s
end{equation}

I was thinking so hard how to take advantage of it.

The only idea that I came up with: turning $A$ into a product of a tensor and a matrix - combining $Lambda_1$ and $Lambda_2$ $longrightarrow
Big(
begin{bmatrix}
-1& 2\
-2 & 2\
end{bmatrix} ,
begin{bmatrix}
1& -1\
2 & -1\
end{bmatrix}
Big)
$ - times a matrix $S=begin{bmatrix}
a & a\
b & b\
end{bmatrix}$. Since the product is just $A$, so it obviously has a eigenvalue, but I am left wondering if it is possible and meaningful to have a "eigen-matrix" or " eigen-value" for the tensor? In particular, for this tensor any way the eigen-matrix related to the eigen-vector of $A$?

(the only tensor I have learned is from mathematical method for physics many years ago, so if this is a very silly question or I made some silly mistakes, please forgive me!)

linear-algebra

share|cite|improve this question

edited 13 hours ago

Shing

asked yesterday

ShingShing

283112

$endgroup$

• 
  
  
  

  
  1
  

  
  
  
  
  
  

  
  $begingroup$
  For your general question: since your tensor is the representation of a linear endomorphism of a finite dimensional vector space, there is a natural concept of isomorphism. For your specific matrix: compute its characteristic polynomial.
  $endgroup$
  – Mindlack
  yesterday
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  @Mindlack thanks for the comment (I had calculated the eigenvalues and eigenvectors of $A$)! would you mind elaborating a bit more about isomorphism (either on comment or answer)?
  $endgroup$
  – Shing
  13 hours ago
  
  
  

  
  
  
  

add a comment | 

1

$begingroup$

Short vision of my question:

Is there such a thing as eigen-matrix of a tensor?

---

Long vision of my question:

I was given

begin{equation}
A=begin{bmatrix}
2b-a & a-b\
2b-2a & 2a-b\
end{bmatrix}
end{equation}

to find its eigenvalue and eigenvector, and I have found them with standard method.

However, I observed that the first column vector of $A$ is:

begin{equation}
begin{bmatrix}
2b-a\
2b-2a\
end{bmatrix} =begin{bmatrix}
-1& 2\
-2 & 2\
end{bmatrix}
begin{bmatrix}
a\
b
end{bmatrix}=Lambda_1s
end{equation}

similarly for the second column:

begin{equation}
begin{bmatrix}
a-b\
2a-b\
end{bmatrix} =begin{bmatrix}
1& -1\
2 & -1\
end{bmatrix}
begin{bmatrix}
a\
b
end{bmatrix}=Lambda_2s
end{equation}

I was thinking so hard how to take advantage of it.

The only idea that I came up with: turning $A$ into a product of a tensor and a matrix - combining $Lambda_1$ and $Lambda_2$ $longrightarrow
Big(
begin{bmatrix}
-1& 2\
-2 & 2\
end{bmatrix} ,
begin{bmatrix}
1& -1\
2 & -1\
end{bmatrix}
Big)
$ - times a matrix $S=begin{bmatrix}
a & a\
b & b\
end{bmatrix}$. Since the product is just $A$, so it obviously has a eigenvalue, but I am left wondering if it is possible and meaningful to have a "eigen-matrix" or " eigen-value" for the tensor? In particular, for this tensor any way the eigen-matrix related to the eigen-vector of $A$?

(the only tensor I have learned is from mathematical method for physics many years ago, so if this is a very silly question or I made some silly mistakes, please forgive me!)

linear-algebra

share|cite|improve this question

edited 13 hours ago

Shing

asked yesterday

ShingShing

283112

$endgroup$

• 
  
  
  

  
  1
  

  
  
  
  
  
  

  
  $begingroup$
  For your general question: since your tensor is the representation of a linear endomorphism of a finite dimensional vector space, there is a natural concept of isomorphism. For your specific matrix: compute its characteristic polynomial.
  $endgroup$
  – Mindlack
  yesterday
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  @Mindlack thanks for the comment (I had calculated the eigenvalues and eigenvectors of $A$)! would you mind elaborating a bit more about isomorphism (either on comment or answer)?
  $endgroup$
  – Shing
  13 hours ago
  
  
  

  
  
  
  

add a comment | 

$begingroup$

Short vision of my question:

Is there such a thing as eigen-matrix of a tensor?

---

Long vision of my question:

I was given

begin{equation}
A=begin{bmatrix}
2b-a & a-b\
2b-2a & 2a-b\
end{bmatrix}
end{equation}

to find its eigenvalue and eigenvector, and I have found them with standard method.

However, I observed that the first column vector of $A$ is:

begin{equation}
begin{bmatrix}
2b-a\
2b-2a\
end{bmatrix} =begin{bmatrix}
-1& 2\
-2 & 2\
end{bmatrix}
begin{bmatrix}
a\
b
end{bmatrix}=Lambda_1s
end{equation}

similarly for the second column:

begin{equation}
begin{bmatrix}
a-b\
2a-b\
end{bmatrix} =begin{bmatrix}
1& -1\
2 & -1\
end{bmatrix}
begin{bmatrix}
a\
b
end{bmatrix}=Lambda_2s
end{equation}

I was thinking so hard how to take advantage of it.

The only idea that I came up with: turning $A$ into a product of a tensor and a matrix - combining $Lambda_1$ and $Lambda_2$ $longrightarrow
Big(
begin{bmatrix}
-1& 2\
-2 & 2\
end{bmatrix} ,
begin{bmatrix}
1& -1\
2 & -1\
end{bmatrix}
Big)
$ - times a matrix $S=begin{bmatrix}
a & a\
b & b\
end{bmatrix}$. Since the product is just $A$, so it obviously has a eigenvalue, but I am left wondering if it is possible and meaningful to have a "eigen-matrix" or " eigen-value" for the tensor? In particular, for this tensor any way the eigen-matrix related to the eigen-vector of $A$?

(the only tensor I have learned is from mathematical method for physics many years ago, so if this is a very silly question or I made some silly mistakes, please forgive me!)

linear-algebra

share|cite|improve this question

edited 13 hours ago

Shing

asked yesterday

ShingShing

283112

$endgroup$

share|cite|improve this question

edited 13 hours ago

Shing

asked yesterday

ShingShing

283112

share|cite|improve this question

edited 13 hours ago

Shing

asked yesterday

ShingShing

283112

asked yesterday

ShingShing

283112

asked yesterday

ShingShing

283112