Finite sum of eigenspaces (with distinct eigenvalues) is a direct sum

Question

I'm trying to show that a finite sum of eigenspaces (with distinct eigenvalues) is a direct sum.

I have $ \alpha : V \to V $. The eigenspaces are $ V_{\lambda_i} = \ker(\alpha - \lambda_i id_V )$ for $ 1 \leq i \leq n $. My attempt at a proof:

$ A + B $ is a direct sum iff $ A \cap B = \{0\} $. If $ v \neq 0 \in V_{\lambda_i} \cap V_{\lambda_j} $ for some $i,j, i \neq j $, then $ \alpha(v) = \lambda_i v $ and $ \alpha(v) = \lambda_j v $. So $(\lambda_i - \lambda_j)v = 0 $, and so $ \lambda_i = \lambda_j $. This is a contradiction, so any pair of the eigenspaces have trivial intersection. Therefore $ \cap_{i=1}^n V_{\lambda_i} = \{0\} $, and so we have a direct sum.

Is this ok?

Thanks

Answer 1

No, this is not a full proof. It is not true that, if $V = A+B+C$, and $A \cap B = A \cap C = B \cap C = \{ 0 \}$, then $V = A \oplus B \oplus C$. For example, let $V = \mathbb{C}^2$ and let $A$, $B$ and $C$ be the one dimensional subspaces spanned by $(1,0)$, $(1,1)$ and $(0,1)$.

This does give some good intuition for why the claim is true. If you want to build your way to the full proof, you might try the special case of three eigenspaces and see what you can do.

Amusingly, this is currently the top voted example of a common false belief over at MO.

Answer 2

To clarify some concepts, I list some relevant definitions from Artin's "Algebra". Suppose $\{W_i\}$ $i=1,\ldots,n$ are vector subspaces of $V$, then their sum is given by $$ W_1 + \cdots + W_n := \{v\in V\ |\ v = w_1 + \cdots + w_n \mbox{ ,with } w_i\in W_i\} $$ The subspaces $W_1,\ldots,W_n$ are independent if $$ w_1 + \cdots + w_n = 0 \mbox{ ,with } w_i\in W_i \mbox{ implies } w_i = 0 \mbox{ for all i} $$ A subspace $U$ is called a direct sum of $W_1,\ldots,W_n$ if $U = W_1 + \cdots + W_n$ and $W_1,\ldots,W_n$ are independent.

From the above definitions, what we real need to show in this problem is that eigenspaces corresponding to distinct eigenvalues are independent.

Let $v_1,\ldots,v_n$ be eigenvectors with eigenvalues $\lambda_1,\ldots,\lambda_n$, respectively, and $v_1 + \cdots + v_n = 0$.

Now, we use induction to show all $v_i = 0$. If $n = 1$, it's trivial. Otherwise, we have the following two observations. If we multiply both sides of the equation by $\lambda_1$, we get $$ \lambda_1 v_1 + \cdots + \lambda_1 v_n = 0 $$ If we apply the linear map $\alpha$ to both sides of the equation, we get $$ \lambda_1 v_1 + \cdots + \lambda_n v_n = 0 $$ Then, we subtract the first equation from the second one $$ (\lambda_2 - \lambda_1) v_2 + \cdots + (\lambda_n - \lambda_1) v_n = 0 $$ Since $\lambda_1,\ldots,\lambda_n$ are distinct, the coefficients in the equation above are non-zero.

By induction $v_1 = \cdots = v_n = 0$.

Hence, eigenspaces corresponding to distinct eigenvalues are independent.

Answer 3

For the record, here is one way to prove that any finite sum of eigenspaces for distinct eigenvalues is a direct sum. (One may in fact drop the "finite", since an infinite direct sum of subspaces is direct if and only if all sums of a finite subset of those subspaces are direct.)

Using induction on the number $n$ of eigenvalues involved (the cases $n\leq1$ are trivial), one may suppose that the sum of $V_{\lambda_1},\ldots,V_{\lambda_{n-1}}$ is already known to be direct. Then it suffices to show that the sum of $V_{\lambda_1}\oplus\cdots\oplus V_{\lambda_{n-1}}$ and $V_{\lambda_n}$ is direct (in this form one can conclude directness of a large sum of subspaces by considering only two subspaces at a time!). For this we may show that $(V_{\lambda_1}\oplus\cdots\oplus V_{\lambda_{n-1}})\cap V_{\lambda_n} = \{0\}$. But this amounts to showing that $\lambda_n$ is not an eigenvalue of the restriction of our linear operator $\alpha$ to the direct sum $V_{\lambda_1}\oplus\cdots\oplus V_{\lambda_{n-1}}$; this is obvious since on a basis of eigenvectors this restriction has a diagonal matrix with all diagonal entries among $\{\lambda_1,\ldots,\lambda_{n-1}\}$. Or equivalently, $\alpha-\lambda_nI$ acts as an invertible operator on each of the summands $V_{\lambda_i}$ (with $i<n$), and therefore also on their sum (combine the inverses).