diagonal matrix help

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May 2nd 2013, 08:31 AM
Stormey

diagonal matrix help

Hi guys.
I have the following matrix:

$\begin{bmatrix}1 & 1 & 0 & 0\\ -1 & -1 & 0 & 0\\ -2 & -2 & 2 & 1\\ 1 & 1 & -1 & 0\end{bmatrix}$

its characteristic polynomial is: $f(\lambda)=\lambda^2(\lambda-1)^2$

I've been asked if the above matrix has diagonal matrix over $\mathbb{R}$ , I wrote "no, since the geometric multiplicity of the eigenvalue 1 is 1 (and thus smaller then the algebraic multiplicity).

and then, I've been asked if the above matrix has diagonal matrix over $\mathbb{C}$ .
I'm guessing no, but I came here to ask, just to make sure I'm not missing anything here:
why does it make a difference if we're over $\mathbb{R}$ or $\mathbb{C}$ in this case?
Is it possible to get a different geometric multiplicity in different fields, for the same roots?
May 2nd 2013, 11:02 AM
Shakarri

Re: diagonal matrix help

You cannot diagonalize the matrix because the 2nd row is a scalar multiple of the first row so there is no way to reduce the first and second row to diagonal form.
May 2nd 2013, 11:37 AM
Stormey

Re: diagonal matrix help

Hi Shakarri, thanks for the help.
I know that I cannot diagnolize it, but that's not what I'm getting at here.
The question is, does it make a difference if A is over $\mathbb{R}$ or over $\mathbb{C}$ , when I need to determine whether or not A has a similar diagonal matrix?

BTW, are you saying that every matrix A such that one of its rows is a scalar multiple of another cannot be diagonalized?
May 2nd 2013, 12:12 PM
Shakarri

Re: diagonal matrix help

Quote:

Originally Posted by Stormey

BTW, are you saying that every matrix A such that one of its rows is a scalar multiple of another cannot be diagonalized?

Yes any matrix with rows that are scalar multiples of others cannot be diagonalized. If you could diagonalize a 4x4 matrix like this then you would be able to use 3 simultaneous equations to solve for 4 variables.
May 2nd 2013, 02:19 PM
Stormey

Re: diagonal matrix help

OK, that's confusing.

take for example the following transformation:
$T(x,y)=T(\frac{x+y}{2},\frac{x+y}{2})$
T could be represnted by:
$A=[T]_E=\begin{bmatrix}\frac{1}{2} & \frac{1}{2}\\ \frac{1}{2} & \frac{1}{2}\end{bmatrix}$
and then, according to what you said, A cannot be diagonalized, since the first row is a scalar multiple of the second, but:

$\begin{bmatrix}1 & 1\\ 1 & -1\end{bmatrix}\begin{bmatrix}\frac{1}{2} & \frac{1}{2}\\ \frac{1}{2} & \frac{1}{2}\end{bmatrix}\begin{bmatrix}\frac{1}{2} & \frac{1}{2}\\ \frac{1}{2} & -\frac{1}{2}\end{bmatrix}=\begin{bmatrix}1 & 0\\ 0 & 0\end{bmatrix}$

so there exist P such that $D=PAP^{-1}$
(D is a matrix in which the entries outside the main diagonal are all zero thus by definition diagonal), and therfore A can be diagonalized.
May 2nd 2013, 06:08 PM
Hartlw

Algebraic and Geometric Multiplicity

Algebraic multipicity is number of identical roots of characteristic equation. Geometric multiplicity is the number of independent eigenvectors corresponding to any multiple root. If algebraic multiplicity equals geometric multiplicity, A can be diagonalized.

In the OP 0 is a root of algebraic multiplicity 2 and 1 is a root of algebraic multiplicity 2

If algebraic multiplicity is k, geometric multiplicity is k if

Rank | A-Iλ_k | is n-k (Mirsky, pg 204),

or just try to find k independent eigenvectors corresponding to λ_k. If, for example, k=2 and you can only find one independent eigenvector, geometric multiplicity is less than algebraic multiplicity and you can’t diagonalize A.
May 3rd 2013, 01:22 AM
Stormey

Re: diagonal matrix help

again, I know all that.
I know what algebraic and geometric multiplicity are.
that was not what I asked.

Quote:

The question is, does it make a difference if A is over \mathbb{R} or over \mathbb{C}, when I need to determine whether or not A has a similar diagonal matrix?
May 3rd 2013, 05:39 AM
Hartlw

Re: diagonal matrix help

Yes. You might not get all the roots in R.

In general, a polynomial can only be factored into factors of the form (x-a) if you allow complex roots. There are complex eigen vectors corresponding to complex roots so rule about algebraic and geometric multiplicity remains the same, ie, in general only over C.
May 3rd 2013, 05:49 AM
HallsofIvy

Re: diagonal matrix help

Quote:

Originally Posted by Shakarri

You cannot diagonalize the matrix because the 2nd row is a scalar multiple of the first row so there is no way to reduce the first and second row to diagonal form.

That has nothing to do with "diagonalizing". The fact that one row is a scalar multiple of another means that the matrix is not invertible, so has determinant 0 and so has 0 as an eigenvalue. Which we already knew.

Stormey, a matrix that has all real eigenvalues is or is not "diagonalizable" over the real number or complex numbers simultaneously. That is if it is not diagonalizable over the real numbers, it is not diagonalizable over the complex numbers.
May 3rd 2013, 05:51 AM
HallsofIvy

Re: diagonal matrix help

Quote:

Originally Posted by Shakarri

Yes any matrix with rows that are scalar multiples of others cannot be diagonalized. If you could diagonalize a 4x4 matrix like this then you would be able to use 3 simultaneous equations to solve for 4 variables.

I think you are confusing "diagonalizable" with "invertible".
May 3rd 2013, 06:07 AM
Hartlw

Re: diagonal matrix help

Quote:

Originally Posted by HallsofIvy

That has nothing to do with "diagonalizing".

Stormey, a matrix that has all real eigenvalues is or is not "diagonalizable" over the real number or complex numbers simultaneously. That is if it is not diagonalizable over the real numbers, it is not diagonalizable over the complex numbers.

Not true. If the characteristic equation is (x-a)(x²+bx+c) =0 in R, where the quadratic has no real roots, the matrix can’t be diagonalized. However, over C you can write (x-a)(x-b)(x-c) = 0 and the matrix can be diagonalized in C.
May 3rd 2013, 06:25 AM
Stormey

Re: diagonal matrix help

Quote:

Originally Posted by HallsofIvy

Stormey, a matrix that has all real eigenvalues is or is not "diagonalizable" over the real number or complex numbers simultaneously. That is if it is not diagonalizable over the real numbers, it is not diagonalizable over the complex numbers.

so in this case (where the cp is $\lambda^2(\lambda-1)^2$ ), since A is not diagonalizable over $\mathbb{R}$ , it is also not diagonalizable over $\mathbb{C}$ .
thank you!
May 3rd 2013, 06:29 AM
Stormey

Re: diagonal matrix help

Quote:

Originally Posted by Hartlw

Not true. If the characteristic equation is (x-a)(x²+bx+c) =0 in R, where the quadratic has no real roots, the matrix can’t be diagonalized. However, over C you can write (x-a)(x-b)(x-c) = 0 and the matrix can be diagonalized in C.

I think it's true for polynomial that can be factorized to a linear factors, where each of the factors is in $\mathbb{R}$ .
May 3rd 2013, 07:14 AM
Hartlw

Re: diagonal matrix help

Quote:

Originally Posted by Stormey

I think it's true for polynomial that can be factorized to a linear factors, where each of the factors is in $\mathbb{R}$ .

That is not the (obvious) situation I was referring to.

I was responding to: "That is if it is not diagonalizable over the real numbers, it is not diagonalizable over the complex numbers," which is not true. (underlining added).
See my previous post.
May 3rd 2013, 07:21 AM
HallsofIvy

Re: diagonal matrix help

Quote:

Originally Posted by Hartlw

Not true. If the characteristic equation is (x-a)(x²+bx+c) =0 in R, where the quadratic has no real roots, the matrix can’t be diagonalized. However, over C you can write (x-a)(x-b)(x-c) = 0 and the matrix can be diagonalized in C.

In that case, the matrix does not have "all real eigenvalues" as I required before.

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