How do I sum the dimensions of a space?

Question

From the wikipedia page on diagonalization

An $n\;x\;n$ matrix $A$ A over a field $F$ is diagonalizable if and only if the sum of the dimensions of its eigenspaces is equal to $n$

My question is, how do I sum the dimensions of the eigenspace or any space for that matter..

Say my column space of a $3x3$ matrix is...

$$ \begin{bmatrix} 1 \\ -1 \\ 0 \end{bmatrix} \begin{bmatrix} 0 \\ -1 \\ 1 \end{bmatrix} $$

How do I "sum the dimensions?"

Answer 1

The set of vectors corresponding to each eigenvalue forms a subspace. You are to take the dimension of each of these subspaces and add them up to see if it adds to the dimension of the whole space. For example, $$\begin {pmatrix} 2&0&0\\0&2&0\\0&0&1 \end {pmatrix}$$ has eigenvalues of $1,2$. The space corresponding to $1$ is vectors proportional to $(0,0,1)^T$. The space corresponding to $2$ is the one spanned by $(1,0,0)^T$ and $(0,1,0)^T$. The sum of the dimensions of these subspaces is $1+2=3$, which is the dimension of the whole space. The matrix is diagonalizable, as can be seen because it is diagonal.

By contrast, $$\begin {pmatrix} 2&1&0\\0&2&0\\0&0&1 \end {pmatrix}$$ has eigenvalues of $1,2$. The space corresponding to $1$ is vectors proportional to $(0,0,1)^T$. The space corresponding to $2$ is the one spanned by$(1,0,0)^T$. The sum of the dimensions of these subspaces is $1+1=2$, which is less than the dimension of the whole space. The matrix is not diagonalizable. I haven't proven that, but the theorem promises it.

Answer 2

Just to add to @RossMillikan's excellent answer ...

In general, all we can say about the dimension of the sum of two vector subspaces $V$ and $W$ is that it is greater than or equal to $\max\{\dim V, \dim W\}$ and less than or equal to $\dim V + \dim W$.

However, if $V$ and $W$ are eigenspaces with distinct eigenvalues $\lambda_V$ and $\lambda_W$ then we know that the only vector they have in common is $0$ (can you prove this ?) and so we can say that

$\dim(V+W) = \dim V + \dim W$

Different eigenspaces have different eigenvalues, so this is why we can simply add the dimensions of eigenspaces to find the dimension of their sum.