# Find Basis for diagonal matrix

1. Oct 27, 2012

### Clandry

I'm not sure how to start this problem.
All i know is a diagonal matrix consists of all 0 elements except along the main diagonal.

But how do I even find a basis for this?

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2. Oct 27, 2012

### haruspex

What would a basis look like? It would be set of nxn matrices such that... you can do what with them?

3. Oct 27, 2012

### Clandry

For this case, a basis consists of all matrices such that all nxn diagonal matrices can be written as a linear combination of them?

4. Oct 27, 2012

### haruspex

Yes. What's the simplest matrix you can think of that might be useful in creating such a basis?

5. Oct 27, 2012

### Clandry

This is where I get stuck. I've only been taught and done problems where the basis is a set of "vectors."

I saw somewhere that the basis for a 2x2matrix is
1 0
0 0

0 1
0 0

0 0
1 0

0 0
0 1

if it were a 2x2 diagonal would it be
1 0
0 0

and
0 0
0 1
?

6. Oct 27, 2012

### haruspex

Ok. Now try 3x3.

7. Oct 27, 2012

### Clandry

1 0 0
0 0 0
0 0 0

0 0 0
0 1 0
0 0 0

0 0 0
0 0 0
0 0 1

if it's an nxn matrix, wouldn't that give an infinite amount of matrices for the bases?

The answer in the back of the book is
37. B = {eii | 1 ≤ i ≤ n} the "ii" part is supposed to be subscripts for e. I'm bad at interpreting these kind of answers, what is it saying?

8. Oct 27, 2012

### haruspex

You had 2 for 2x2 and 3 for 3x3. Why would you get infinitely many for nxn?

9. Oct 27, 2012

### Clandry

oops i mean n amount.

10. Oct 27, 2012