Diagonalizing A: Is it Possible with Rational Numbers?

[quasar987]

Hello matrix gurus,

Is is true that if A is real with A²=I (eigenvalues ±1), it is diagonalizable over R?

What if I add that A is in O(m,m), where O(m,m) is the split indefinite orthogonal group of 2m x 2m matrices M such that M^TI_{m,m}M=I_{m,m}, where I_{m,m} is the block diagonal matrix diag(I_m,-I_m) for I_m the m x m identity matrix?

Thanks

 

[micromass]

Yes, an involution is always diagonalizable over the reals.

We use the following result:

Another characterization: A matrix or linear map is diagonalizable over the field F if and only if its minimal polynomial is a product of distinct linear factors over F. (Put in another way, a matrix is diagonalizable if and only if all of its elementary divisors are linear.)

See http://en.wikipedia.org/wiki/Diagonalizable_matrix A proof can be found in Hoffman & Kunze theorem 6 page 204.

The minimal polynomial of an involution is x^2-1=(x+1)(x-1) and satisfies our theorem.

 

[lavinia]

Hello matrix gurus,

Is is true that if A is real with A²=I (eigenvalues ±1), it is diagonalizable over R?

What if I add that A is in O(m,m), where O(m,m) is the split indefinite orthogonal group of 2m x 2m matrices M such that M^TI_{m,m}M=I_{m,m}, where I_{m,m} is the block diagonal matrix diag(I_m,-I_m) for I_m the m x m identity matrix?

Thanks

A.v - v and A.v + v are both eigen vectors of A. They can not both be zero.
Use this inductively to find a basis of the vector space made up of eigen vectors of A.

e.g.
If A.v is not a multiple of v then both A.v - v and A.v + v are non-zero eigen vectors. Choose any other vector that is not in the span of v and A.v. and continue.

One certainly does not need all of the reals to diagonalize A. The rationals will work. I think all you really need to be able do is divide by 2.

Over the integers A can not always be diagonalized. The map (1,0) -> (1,-1) (0,1) -> (0,-1) is an example. The problem of inequivalent integer representations of groups is daunting.