# matrix canonical form

• October 29th 2009, 03:54 AM
charikaar
matrix canonical form
let A= $\begin{bmatrix}6 & 3 \\ 1 & 6\end{bmatrix}$

be a matrix over the field
K = F_11(the field of integers mod p). What is the rank of A, and what is its canonical form for equivalence? Briefly justify your answer.

Rank of A is 2 because there are two linearly independant rows or columns. How do i start solving the other part? what is its canonical form for equivalence?

Thank you.
• October 29th 2009, 04:43 AM
tonio
Quote:

Originally Posted by charikaar
let A= $\begin{bmatrix}6 & 3 \\ 1 & 6\end{bmatrix}$

be a matrix over the field
K = F_11(the field of integers mod p). What is the rank of A, and what is its canonical form for equivalence? Briefly justify your answer.

Rank of A is 2 because there are two linearly independant rows or columns. How do i start solving the other part? what is its canonical form for equivalence?

Thank you.

Nop, it's rank cannot be two since its determinant is zero...
If you now evaluate its characteristic polynomial you'll find the matrix has two different eigenvalues and thus its Jordan Canonical form is...

Tonio
• October 29th 2009, 04:48 AM
charikaar
Quote:

Originally Posted by tonio
Nop, it's rank cannot be two since its determinant is zero...
If you now evaluate its characteristic polynomial you'll find the matrix has two different eigenvalues and thus its Jordan Canonical form is...

Tonio

Isn't det(A)=36-3=33? Do i have to do anything with field K=F_11?

thanks
• October 29th 2009, 05:04 AM
tonio
Quote:

Originally Posted by charikaar
Isn't det(A)=36-3=33? Do i have to do anything with field K=F_11?

thanks

Of course you have to do "anything" with the field $\mathbb{F}_{11}$ : it is the field of definition of your matrix and thus you have to work out ALL the operations (mod 11)! Including the determinant.

Tonio
• October 29th 2009, 09:37 AM
charikaar
I know there are $11^4$ matrices in http://www.mathhelpforum.com/math-he...9d73319f-1.gif and elements of http://www.mathhelpforum.com/math-he...9d73319f-1.gif are 1,2,4,8,5,10,9,7,3,6.

Can you help me a bit further as I still can't solve the problem.

thanks