Thread: Transposition of Matrix as a Linear Transformation - Find e-values, e-vectors

1. Transposition of Matrix as a Linear Transformation - Find e-values, e-vectors

We define a linear transformation $\displaystyle T:Mat_{m,m}->Mat_{m,m}$ of $\displaystyle m\times m$ matrices such that $\displaystyle T(M)=M^{Tr}$.

Basically we are looking at the transposition of a matrix, treating it as a linear transformation. I need to find characteristic polynomials, eigenvalues, eigenvectors, etc. I started by looking at 2-by-2 and 3-by-3 to get some insight. I attached the calculations.

The 2x2 and 3x3 helped a little, but I still cannot figure out how to generalize it.

2. Re: Transposition of Matrix as a Linear Transformation - Find e-values, e-vectors

$\displaystyle T^2=I$

3. Re: Transposition of Matrix as a Linear Transformation - Find e-values, e-vectors

Originally Posted by Idea
$\displaystyle T^2=I$
Can you explain a bit? I have read that any matrix such that $\displaystyle A^2=I$ is diagonalizable. I'm not clear how to apply that here.

4. Re: Transposition of Matrix as a Linear Transformation - Find e-values, e-vectors

the minimal polynomial is $x^2-1$

so the eigenvalues are $\displaystyle \{-1,1\}$

yes it is diagonalizable

the characteristic polynomial is of the form $\displaystyle (x+1)^a(x-1)^b$

5. Re: Transposition of Matrix as a Linear Transformation - Find e-values, e-vectors

Originally Posted by Idea
the minimal polynomial is $x^2-1$

so the eigenvalues are $\displaystyle \{-1,1\}$

yes it is diagonalizable

the characteristic polynomial is of the form $\displaystyle (x+1)^a(x-1)^b$
Can you explain you obtained that? I'll try to reproduce your solns but I'm not sure how to start.

6. Re: Transposition of Matrix as a Linear Transformation - Find e-values, e-vectors

the minimal polynomial of $T$ is the monic polynomial of least degree that $T$ satisfies

now $T^2(M)=T(M^t)=(M^t)^t=M$ so $T$ satisfies $x^2-1$

It is clear that $T$ does not satisfy any non zero polynomial of degree less than $2$

Therefore $x^2-1$ is the minimal polynomial of $T$