How can I construct 2 x 2 normal matrices A and B such that A + B is not normal?

[tazzzdo]

Normal Matrices Examples (URGENT)

I need to produce 2 x 2 normal matrices A and B such that A + B is not normal. I have proven that AB is normal if AB = BA using the Householder matrix form. But I can't find a form for A + B failing to be normal.

A matrix A with entries a, b, c, d is going to be normal if b = c. But that means a matrix B has to have b = c to be normal as well. So A + B will have 2b, 2c for the 1,2 and 2,1 entries as well which will make A + B normal.

Unless I'm totally off. I have to get this done in the next 3 hours.

 

[micromass]

Please don't write things like

I have to get this done in the next 3 hours.

or "urgent". You will actually get less replies that way.

 

[berkeman]

Schoolwork?

 

[mathwonk]

relax, the world will not end if this drags out.

I suggest starting by thinking up a really simple non normal matrix, then try to realize it as A+B where A and B are a little nicer. (This took me about 30 seconds.)

 

[micromass]

Schoolwork?

I've moved it to homework.

 

[tazzzdo]

I don't understand how to make a non-normal matrix break down into 2 normal matrices. That's the whole point of my post.

 

[micromass]

I don't understand how to make a non-normal matrix break down into 2 normal matrices. That's the whole point of my post.

Can you give us an example of a non-normal matrix?

 

[tazzzdo]

1 1
0 1

There is no way to make that matrix the sum of 2 normal matrices A and B?

 

[Dick]

1 1
0 1

There is no way to make that matrix the sum of 2 normal matrices A and B?

Sure there's a way. Try and write it as the sum of a symmetric matrix and an antisymmetric matrix.

 

[tazzzdo]

Now I understand. I used

0 1
1 0

and

0 -1
1 0

 

[Dick]

Now I understand. I used

0 1
1 0

and

0 -1
1 0

Sure. That works.

 

[tazzzdo]

Love being in grad school and missing stupid things like that. -__- Preciate the help.

 

[mathwonk]

nice hint, micro mass. good job, tazzzdo.