Is {C, D} Independent? Proving Matrix Independence in 3 Simple Steps

[jumbogala]

Edit: There was a mistake in the question, see below for right question.

 

[Mandark]

If {C,D} was not a set of independent matrices then what sort of relation would C and D satisfy? What does taking the transpose of both sides of said relation tell you?

 

[jumbogala]

If {C,D} were not a set of independent matrices, then I would be able to find some numbers a and b not both zero, such that aC + bD = 0.

Transpose of this... aC^T + bD^T = 0?

 

[Mandark]

Right, now use the relations given in the question to rewrite that last equation.

 

[jumbogala]

Okay, so that means

aC + b(-D) = 0.

Now that means that aC + b(-D) = aC + bD

and can be rewritten aC -b(D) = aC + bD... which would mean that a = a, and -b = b. But the only way -b = b can be true is if b = 0. So therefore, b is zero and my initial condition is false, and they are independent! right?

 

[jumbogala]

oh wait, but a can still be something other than zero... my condition is that they are not BOTH zero. so my initial condition is not necessarily false =/

 

[Mandark]

Sort of. You can't exactly just equate coefficients if that's what you did. The last equation you wrote implies that bD = 0 by just rearranging. You can't cancel the D here either, but you can use one property given in the question which you haven't used yet. :)

 

[jumbogala]

Wait, why can't I cancel the D? If bD = 0 and D is not zero, then isn't b zero?

Am I supposed to use the fact that they are n x n?

 

[jumbogala]

I just realized the question is not supposed to say that C and D are independent matrices, just that they are n x n and nonzero, if that makes a difference.

 

[Dick]

Sure it makes a difference. You've got aC+bD=0 and aC-bD=0. To show C and D are independent, you want to show that a and b are both zero. Suppose a is nonzero?

 

[jumbogala]

If it's still true that bD is zero, then if a is nonzero, you'd have aC is not zero.

Then aC + bD = 0 --> aC + 0 = 0 but if aC is not zero this is obviously not true.

 

[Dick]

Add the two equations. Who cares what bD is? If a is nonzero then you get 2aC=0. a is nonzero, C is nonzero. Possible, or not?

 

[jumbogala]

No, not possible. Therefore a must be zero.

But I still need to show that b is zero, right?

Now that I know a is zero can I just plug it into the equation aC + bD = 0?

 

[Dick]

Sure you can. Or you could subtract the two equations and conclude 2bD=0, D being nonzero means b must be zero. Same conclusion either way.

 

[jumbogala]

Great, so then a = b = 0, thus {C, D} is independent. And we're done!

Thanks to both of you for your help :)