What is the nullity of a zero matrix?

[SprucerMoose]

Gday,

I was wondering if someone could tell what the nullity of an nxn zero matrix is? I can't decide if its zero or n. Could someone knowledgeable please enlighten me?Thanks

 

[micromass]

Hi SprucerMoose

Well, the nullity of a matrix is defined as the dimension of it's nullspace (or kernel). So let O be our zero matrix, then the nullspace is

$$\{x\in \mathbb{R}^n~\vert~Ox=0\}$$

Clearly, every vector satisfies Ox=0. Thus the nullspace is entire $$\mathbb{R}^n$$. The dimension of $$\mathbb{R}^n$$ is n. Hence, the nullity of the zero matrix is n.

Please note, that the matrices with nullity 0 are exactly the invertible matrices (in finite-dimensional spaces of course). And, as you might know, the zero matrix is far from being invertible!

 

[SprucerMoose]

Thanks very much for the speedy and clear response.