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# eigenspace

Let $V$ be a vector space over a field $k$. Fix a linear transformation $T$ on $V$. Suppose $\lambda$ is an eigenvalue of $T$. The set $\{v\in V\mid Tv=\lambda v\}$ is called the *eigenspace* (of $T$) corresponding to $\lambda$. Let us write this set $W_{{\lambda}}$.

Below are some basic properties of eigenspaces.

1. 2. The dimension of $W_{{\lambda}}$ is called the geometric multiplicity of $\lambda$. Let us denote this by $g_{{\lambda}}$. It is easy to see that $1\leq g_{{\lambda}}$, since the existence of an eigenvalue means the existence of a non-zero eigenvector corresponding to the eigenvalue.

3. $W_{{\lambda}}$ is an invariant subspace under $T$ ($T$-invariant).

4. $W_{{\lambda_{1}}}\cap W_{{\lambda_{2}}}=0$ iff $\lambda_{1}\neq\lambda_{2}$.

5. In fact, if $W_{{\lambda}}^{{\prime}}$ is the sum of eigenspaces corresponding to eigenvalues of $T$ other than $\lambda$, then $W_{{\lambda}}\cap W_{{\lambda}}^{{\prime}}=0$.

From now on, we assume $V$ finite-dimensional.

Let $S_{T}$ be the set of all eigenvalues of $T$ and let $W=\oplus_{{\lambda\in S}}W_{{\lambda}}$. We have the following properties:

1. If $m_{{\lambda}}$ is the algebraic multiplicity of $\lambda$, then $g_{{\lambda}}\leq m_{{\lambda}}$.

2. Suppose the characteristic polynomial $p_{T}(x)$ of $T$ can be factored into linear terms, then $T$ is diagonalizable iff $m_{{\lambda}}=g_{{\lambda}}$ for every $\lambda\in S_{T}$.

3. In other words, if $p_{T}(x)$ splits over $k$, then $T$ is diagonalizable iff $V=W$.

For example, let $T:\mathbb{R}^{2}\to\mathbb{R}^{2}$ be given by $T(x,y)=(x,x+y)$. Using the standard basis, $T$ is represented by the matrix

$M_{T}=\begin{pmatrix}1&1\\ 0&1\end{pmatrix}.$

From this matrix, it is easy to see that $p_{T}(x)=(x-1)^{2}$ is the characteristic polynomial of $T$ and $1$ is the only eigenvalue of $T$ with $m_{1}=2$. Also, it is not hard to see that $T(x,y)=(x,y)$ only when $y=0$. So $W_{1}$ is a one-dimensional subspace of $\mathbb{R}^{2}$ generated by $(1,0)$. As a result, $T$ is not diagonalizable.

## Mathematics Subject Classification

15A18*no label found*

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