i would like to clarify everything about Frobenius normal form,so please help me to understand it .as i read from wikipedia
In linear algebra, the Frobenius normal form, Turner binormal projective form or rational canonical form of a square matrix A is a canonical form for matrices that reflects the structure of the minimal polynomial of A and provides a means of detecting whether another matrix B is similar to A without extending the base field F it is basic definition of this term.also we know that two matrix are similar to each other,if they satistfy following form
In linear algebra, two n-by-n matrices A and B are called similar if
$B=P^{-1}* A *P$
for some invertible matrix $P$
so does it mean we should find such matrix which has similar structure as given mattrix and also minimum polynomial? so for example we given matrix
B=[3 4; 2 5]
B =
3 4
2 5
it's characteristic equation is
poly(B)
ans =
1 -8 7
or $a^2-8*a+7=(a-1)*(a-7)$
what should i do next? as i understand from Wikipedia ,suppose that characteristic equation is $a^2-2*a+1=(a-1)^2$
so it means that first matrix should be such matrix whose characteristic equation should be $a-1$ ?or we have
A=[0 1;1 0]
A =
0 1
1 0
? if this is case,then how many times should i take tensor or Kronecker product or how many copy should i do of given matrix?in copy i meant $A_2=A_1$ $A_3=A_1$ and so on,then take product(tensor product of all this matrix) ? so as i understand because this is second order polynomial,i should take only two matrix?but what if we can't get such kind of form?for example like this we get first time
$a^2-8*a+7=(a-1)*(a-7)$
what we should do in such case?thanks very much,please help me to clarify this things