2n x 2n matrices without real eigenvalues

[pyroknife]

Homework Statement

For an arbitrary positive integer $n$, give a $2n$ x $2n$ matrix $A$ without real eigenvalues.

Homework Equations

The Attempt at a Solution

First of all, I am having some trouble interpreting this problem. I do not know if it is generalized where I am supposed to find a $2n$ x $2n$ matrix A without real eigenvalues for ANY $n$, or just for one specific $n$ value, which I can pick. I assume it is the former.

Since it is 2*n, we know that the number of rows/columns is even for any $n$. If it was only $n$, without it being multiplied it by 2, then the number of rows/columns could be odd, and thus there will always be at least one real eigenvalue.

Since there's an even # of rows/columns, it is possibly that matrix $A$ not have any real eigenvalues.

This is about as far as I got and I am having a very hard time figuring out how to generalize this problem for any arbitrary value of $n$.

 

[LCKurtz]

Homework Statement

For an arbitrary positive integer $n$, give a $2n$ x $2n$ matrix $A$ without real eigenvalues.

Homework Equations

The Attempt at a Solution

First of all, I am having some trouble interpreting this problem. I do not know if it is generalized where I am supposed to find a $2n$ x $2n$ matrix A without real eigenvalues for ANY $n$, or just for one specific $n$ value, which I can pick. I assume it is the former.

The former is correct.

Since it is 2*n, we know that the number of rows/columns is even for any $n$. If it was only $n$, without it being multiplied it by 2, then the number of rows/columns could be odd, and thus there will always be at least one real eigenvalue.

Since there's an even # of rows/columns, it is possibly that matrix $A$ not have any real eigenvalues.

This is about as far as I got and I am having a very hard time figuring out how to generalize this problem for any arbitrary value of $n$.

I would start small. Can you do it for 2x2 matrix? A 4x4? Maybe that will give you some ideas.

 

[pyroknife]

The former is correct.
I would start small. Can you do it for 2x2 matrix? A 4x4? Maybe that will give you some ideas.

2x2 matrix would be simple as the following would be one example
$\begin{bmatrix} 0 & 1\\ -1 & 0 \end{bmatrix}$

I am trying to think of a 4x4.

 

[LCKurtz]

Good so far. Think of that as a building block for your 4x4.

 

[pyroknife]

Good so far. Think of that as a building block for your 4x4.

I think part of the problem I am having is writing down the determinant of $(A-\lambda I)$ for larger systems.

 

[LCKurtz]

Well, lots of zero's helps. If your matrix were diagonal its determinant would be easy, but that won't work. Think in terms of diagonal blocks.

 

[pyroknife]

Well, lots of zero's helps. If your matrix were diagonal its determinant would be easy, but that won't work. Think in terms of diagonal blocks.

Okay let's see...
I'm not too familiar with the term "diagonal blocks."
I assume I can stack the 2 x 2 matrix I previously had along the tri-diagonal of a 4 x 4 matrix such that it looks like:
$\begin{bmatrix} 0 & 1 & 0 & 0\\ -1 & 0 & 0 & 0\\ 0 & 0 & 0 & 1\\ 0 & 0 & -1 & 0 \end{bmatrix}$

I just checked and this has all imaginary eigenvalues.

 

[pyroknife]

Ahh yes, it looks like I can place the original 2x2 matrix I had along the tri-diagonal band of any $2n$ x $2n$ matrix and obtain a matrix with only non-real eigenvalues.

 

[LCKurtz]

Heh heh. Looks like you are on to something, eh? You can probably find some theorems about determinants of matrices such as this. They are called block matrices.

 

[LCKurtz]

I have to hit the sack. I'm guessing you can take it from here. If not, see you tomorrow.

 

[pyroknife]

Heh heh. Looks like you are on to something, eh? You can probably find some theorems about determinants of matrices such as this. They are called block matrices.

I saw this http://math.stackexchange.com/questions/898151/block-matrix-determinant
and the first reply is very elucidating.

 

[pyroknife]

I have to hit the sack. I'm guessing you can take it from here. If not, see you tomorrow.

Thanks! I'm just trying to figure out how to write this succintly.

 

[epenguin]

Could you think of physical simultaneous linear d.e.s that have to give 4 nonreal eigenvalues (vibrational modes) and get their eigenvalue equation, can choose to try find simplest or find most general?

 

[pyroknife]

I have to hit the sack. I'm guessing you can take it from here. If not, see you tomorrow.

Hey LCKurtz, is there some terminology of method to succinctly write these matrices for an arbitrary n?
What I came up with is a bit wordy...

 

[pyroknife]

Could you think of physical simultaneous linear d.e.s that have to give 4 nonreal eigenvalues (vibrational modes) and get their eigenvalue equation, can choose to try find simplest or find most general?

Hmmm, I have not thought about placing this problem in the context of differential equations. I can see how that would help, but I have only just touched the surface in both subjects (linear algebra and differential equations) that connecting the two is bit outside of my knowledge.

 

[LCKurtz]

Hey LCKurtz, is there some terminology of method to succinctly write these matrices for an arbitrary n?
What I came up with is a bit wordy...

Look here:
http://en.wikipedia.org/wiki/Block_matrix#Block_diagonal_matrices

and look at the first picture under Block Diagonal Matrices (3 in the contents). You could probably work with something like that.

 

[pyroknife]

Look here:
http://en.wikipedia.org/wiki/Block_matrix#Block_diagonal_matrices

and look at the first picture under Block Diagonal Matrices (3 in the contents). You could probably work with something like that.

Thanks!

 

[epenguin]

Hmmm, I have not thought about placing this problem in the context of differential equations. I can see how that would help, but I have only just touched the surface in both subjects (linear algebra and differential equations) that connecting the two is bit outside of my knowledge.

A connection to remember to make later. I wonder if linear algebra as "subject" would be on most scientists' curriculum without it?