Similarity transformation to the transpose

In summary, there are certain conditions under which a matrix P exists that transforms a given matrix A to its transpose. This can be found using a linear equation for the coefficients of P. However, there is no general method for finding such a matrix P for any given matrices A and B. The existence of such a matrix also leads to a strange antihomorphism and can be extended to the entire group generated by A and B. Additionally, a matrix P can be found that transforms AT into A and also gives a specific vector c.
  • #1
Leo321
38
0
I have a real nxn matrix A and I want to find P, so that P-1AP=AT. Does such a matrix exist? How do I find it?
What if I have two matrices A,B. Does there exist a matrix P, that transforms both of them to their transposes?
Thanks
 
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  • #2
This isn't exactly what you're asking for, so I don't know if it's useful or not, but if we let X be the n×n matrix with 1 on the diagonal from lower left to upper right and 0 everywhere else, we have XAX=AT.
 
  • #3
Fredrik said:
This isn't exactly what you're asking for, so I don't know if it's useful or not, but if we let X be the n×n matrix with 1 on the diagonal from lower left to upper right and 0 everywhere else, we have XAX=AT.

No, I need X-1AX=AT.
 
  • #4
But it's exactly the same thing. If X is the matrix with ones on the "anti-diagonal" (running from upper right to lower left) then [itex]X= X^{-1}[/itex].
 
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  • #5
Lol, I didn't even realize that myself. :smile: (That's why I said "this isn't exactly what you're asking for").
 
  • #6
But it doesn't work. X with anti-diagonal 1's doesn't give the transpose, it gives a kind of mirroring of the matrix.

I'm also looking for an X such that X^-1AX=A^T. Shouldn't there be a method for generically finding a similarity transformation that gives you a matrix B, whether it's the transpose of A or something else?
 
  • #7
Ah, crap. You're right, if we e.g. consider a 5×5 matrix, the element on row 4, column 1 will end up on row 2, column 5.

I don't know any such method.
 
  • #8
For any particular matrix A, a solution for X can be found.
A X - X A^T = 0 is just a linear equation for the coefficients of X.
Experimentation shows that in general, for a n*n system, only n (n-1) components of X are fixed. So there is some freedom in the choice of X.

Some experimentation also shows that, in general, it's not possible to find an X that transposes two matrices, ie such that
A X = X A^T and B X = X B^T.
I'm not sure what conditions A and B would have to satisfy such that a solution exists.

Note that if it does hold true for A and B, ie
A X = X A^T and B X = X B^T
then you must have a strange antihomorphism
A B X = A X X^{-1} B X = X A^T B^T = X (B A)^T
I'm not sure what that means. But it does extend to the entire group generated by A and B...
 
  • #9
I found the following. For a given matrix A and vectors b,c, this will transform AT into A and also give c = Pb.
[tex]C=\begin{bmatrix} b & Ab & \dots & A^{n-1}b \end{bmatrix}[/tex]
[tex]O=\begin{bmatrix} c & A'c & \dots & (A')^{n-1}c \end{bmatrix}[/tex]
[tex]P= O C ^{-1}[/tex]
 

1. What is a similarity transformation?

A similarity transformation is a mathematical operation that preserves the shape of an object. It involves applying a linear transformation to an object, such as a matrix or a geometric figure, while maintaining its size and orientation.

2. What is the transpose of a matrix?

The transpose of a matrix is a new matrix that is obtained by switching the rows and columns of the original matrix. This means that the elements in the first row of the original matrix become the first column of the transposed matrix, and so on.

3. How is similarity transformation related to the transpose?

Similarity transformation and transpose are related in that a similarity transformation can be applied to a matrix to make it similar to its transpose. This means that the resulting matrices will have the same eigenvalues and eigenvectors, and thus represent the same linear transformation.

4. Why is similarity transformation to the transpose important?

Similarity transformation to the transpose is important in mathematics and science because it allows us to simplify and analyze complex matrices by transforming them into similar matrices that are easier to work with. It also has applications in fields such as physics, engineering, and computer science.

5. How is similarity transformation to the transpose used in real-world applications?

Similarity transformation to the transpose is used in various real-world applications, such as image and signal processing, data compression, and pattern recognition. It is also used in physics to study the behavior of linear systems and in computer graphics to create 3D transformations of objects.

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