A linear operator commuting with all such operators is a scalar multiple of the identity.

Question

The question is from Axler's "Linear Algebra Done Right", which I'm using for self-study.

We are given a linear operator $T$ over a finite dimensional vector space $V$. We have to show that $T$ is a scalar multiple of the identity iff $\forall S \in {\cal L}(V), TS = ST$. Here, ${\cal L}(V)$ denotes the set of all linear operators over $V$.

One direction is easy to prove. If $T$ is a scalar multiple of the identity, then there exists a scalar $a$ such that $Tv = av$, $\forall v \in V$. Hence, given an arbitrary vector $w$, $$TS(w) = T(Sw) = a(Sw) = S(aw) = S(Tw) = ST(w)$$ where the third equality is possible because $S$ is a linear operator. Then, it follows that $TS = ST$, as required.

I am, however, at a loss as to how to tackle the other direction. I thought that a proof by contradiction, ultimately constructing a linear operator $S$ for which $TS \neq ST$, might be the way to go, but haven't made much progress.

Thanks in advance!

Answer 1

For a basis-free answer, consider $S \in L(V)$ given by $S x = f(x) v$ for some vector $v$ and some linear functional $f$ on V. Then $T S x = f(x) T v = S T x = f(T x) v$ for any x. In particular, as long as a nontrivial linear functional $f$ on $V$ exists, there is $x$ such that $f(x) \ne 0$, and then $T v = \alpha v$ for all $v$, where $\alpha = f(T x)/f(x)$. This works even for infinite-dimensional spaces, although I think in general you need the Axiom of Choice to get a nontrivial linear functional on a vector space.

Answer 2

Suppose $TS = ST$ for every $S$. Show that $Tv = a_{v}v$ for every $v\in V$ where $a_v$ could depend on $v$. In other words, show that $v$ and $Tv$ are linearly dependent for each $v \in V$.

Suppose for contradiction that they are linearly independent. Since $(v, Tv)$ is linearly independent, it can be extended to a basis $(v,Tv, u_1, \dots, u_n)$ of $V$. So define $S$ as following: $Sv = v$, $S(Tv) = v$ and $S(u_1) = 0, \dots, S(u_n) = 0$. Then, $Tv = TSv = STv = v$. Hence $v$ and $Tv$ are linearly dependent, which is a contradiction. Then you have to show uniqueness.

Answer 3

In general, when one has a condition of the form "$A$ is a blah if and only if for every $B$ this happens", the "if" direction can often be established by selecting suitably/cleverly chosen $B$ that show everything works.

This is just such a situation.

Let $\beta = \{\mathbf{v}_i\}_{i\in I}$ be a basis for $\mathbf{V}$. For each $i,j\in I$, let $S_{ij}$ be the linear operator on $\mathbf{V}$ given by $$S_{ij}(\mathbf{v}_k) = \left\{\begin{array}{ll} \mathbf{v}_j &\mbox{if $k=i$,}\\ \mathbf{v}_i &\mbox{if $k=j$,}\\ \mathbf{0} &\mbox{if $k\neq i$ and $k\neq j$.} \end{array}\right.$$ That is: for $i\neq j$, $S_{ij}$ exchanges $\mathbf{v}_i$ and $\mathbf{v}_j$, and maps all other basis elements to $\mathbf{0}$. And $S_{ii}$ maps $\mathbf{v}_i$ to itself, and all other basis elements to $\mathbf{0}$. These are our "suitably chosen" $S$.

Now consider $S_{ii}T(\mathbf{v}_j)$ and $TS_{ii}(\mathbf{v}_j)$ first to get information about what $T$ does to $\beta$; then consider $S_{ij}T(\mathbf{v}_j)$ and $TS_{ij}(\mathbf{v}_j)$ for $i\neq j$.

Answer 4

One demonstration of your "other direction":

If $T$ and $S$ are two operators that comute, then $S(\mathrm{Ker}T)\leqslant \mathrm{Ker}T$. In fact, $$v \in \mathrm{Ker}T \Rightarrow T(Sv)=S(Tv)=S(0)=0$$ In words, $\mathrm{Ker}T$ is invariant under $S$.

So, in our case we have that $\mathrm{Ker}T$ is invariant under any linear transformation in $L(V)$. This implies that $\mathrm{Ker}T = V$ or $0$. In fact, in other cases we would have $S\in L(V)$ such that $S(\mathrm{Ker}T)\nsubseteq\mathrm{Ker}T$.

We now show that $T$ has an eigenvalue. In fact, let $S \in L(V)$ be a projection on a non-zero one-dimensional subspace $\langle v \rangle$. Since, $$Tv=T(Sv)=S(Tv)$$ we have that $Tv \in \langle v\rangle$. Equivalently, $Tv=\lambda v$ for some $\lambda \in \mathbb K$.

Since $$TS=ST \iff (T-\lambda I)S=S(T-\lambda I)$$ for any $S \in L(V)$, we have as above that $$\mathrm{Ker}(T-\lambda I)=V$$ or $$\mathrm{Ker}(T-\lambda I)=0.$$ But, since $\lambda$ is an eigenvalue, $\mathrm{Ker}(T-\lambda I)\neq 0$. Therefore, $T=\lambda I$. QED.

This result is a special case of the Schur Lemma wich states: "If $T$ is an operator in $V$ with an eigenvalue $\lambda \in \mathbb K$ and $C \subseteq L(V)$ is a set of operators such that $$TS=ST \forall S \in C$$ and for each non-trivial subspace $W$ there is some $S \in C$ such that $$S(W) \nsubseteq W,$$ then we must have $T=\lambda I$". And whose demonstration is essentialy as above.

Answer 5

Perhaps somewhat against the spirit of Axler's book "linear maps over matrices" (although quite conceptual):

Suppose T commutes with all S. Then in particular it commutes with all invertible S: so $T=STS^{-1}$ for all invertible S. But this means the matrix of T is the same, no matter what basis we choose!

Then it must be diagonal: for fixed $j$, replace basis vector $e_j$ with $2e_j$; then if $i\neq j$, we get $t_{ij}=2t_{ij}$, so $t_{ij}=0$.

Edit[elaboration on the previous line]: Suppose $T$ has matrix $(t_{ij})_{ij}$ w.r.t. the basis $\{e_1,...,e_n\}$. Fix $k$, and consider the basis $B_k=\{v_1,..,v_n\}$ where $v_i=e_i$ if $i\neq k$ and $v_k=2e_k$. Then, for $i\neq k$, the matrix of $T$ w.r.t. $B_k$ has $2t_{ik}$ at entry $i,k$. Hence $t_{ik}=2t_{ik}$ and consequently $2t_{ik}=0$. [End edit.]

Then also all diagonal entries are the same: for fixed $i$ and $j$, interchange $e_j$ and $e_i$, and get $t_{ii}=t_{jj}$.

Answer 6

If $v$ is an eigenvector of T with eigenvalue $\lambda$, then $Sv$ is also an eigenvector with the same eigenvalue. For any two $v$ and $w$ in $V$, there exists a transformation $S$ mapping $v$ to $w$; so all elements of $V$ are eigenvectors with eigenvalue $\lambda$.

Answer 7

Let $\{e_1,\dots,e_n\}$ be a basis for the space $V$. Then you need to show that there exists $a \in \mathbb{R}$ such that $Te_i = a e_i$ for $i = 1,\dots,n$ (every linear operator on finite dimensional space is determined by its values on basis vectors).

Let $S_i \in L(V)$ be defined as $S(a_1 e_1 + \dots + a_n e_n) = a_i e_i$. Assume that $Te_i = b_{i,1} e_1 + \dots + b_{i,n} e_n$. Then $$ T e_i = TS_i e_i = S_i Te_i = b_{i,1} S_i e_1 + \dots + b_{i,n} S_i e_n = b_{i,i} e_i. $$ Now we need to show that $b_{i,i} = b_{j,j}$ for all $i,j=1,\dots,n$. For a given $i,j$ let $S \in L(V)$ be defined by $$ S(a_1 e_1 + \dots + a_n e_n) = a_j e_i + a_i e_j. $$ Then $$ b_{i,i} e_i + b_{j,j} e_j = T(e_i + e_j) = TS(e_i + e_j) = ST(e_i + e_j) = S(b_{i,i} e_i + b_{j,j} e_j) = b_{j,j} e_i + b_{i,i} e_j. $$ Hence, because $(e_k)$ form basis, we obtain $b_{i,i} = b_{j,j}$ which completes the proof.

Answer 8

Multiples of the identity commutate with all other matrices. Now consider some other matrix A defined by $$ A = \begin{bmatrix} a & b \\ c & d \\ \end{bmatrix} $$ We see that $$ \begin{bmatrix} a & b \\ c & d \\ \end{bmatrix} \begin{bmatrix} 1 & 0 \\ 0 & 0 \\ \end{bmatrix} = \begin{bmatrix} a & b \\ 0 & 0 \\ \end{bmatrix}, $$ and $$ \begin{bmatrix} 1 & 0 \\ 0 & 0 \\ \end{bmatrix} \begin{bmatrix} a & b \\ c & d \\ \end{bmatrix} = \begin{bmatrix} a & 0 \\ c & 0 \\ \end{bmatrix}. $$ For these two matrices to be equal we need $c=b=0$. Doing the same trick with $$ \begin{bmatrix} 0 & 1 \\ 1 & 0 \end{bmatrix} $$ will give that a=d. So all matrices that commute with these two matrices are multiples of the identity.

Answer 9

Given a nonzero vector $v \in V$, let $P_v$ denote projection onto the line $\langle v \rangle$. Then, we have $$Tv = T(P_v v) = P_v (Tv)$$ which implies $Tv = \lambda_v v$ for some $\lambda_v \in k$. Note that $$c \lambda_{v} v=c T(v)=T(cv) = \lambda_{cv} cv$$ implies $\lambda_{cv} = \lambda_{v}$ for all $c\in k$ and $v\in V$. Furthermore, we have $$\lambda_{v} v + \lambda_{w} w=Tv + Tw=T(v+w) = \lambda_{v+w} (v+w) \iff (\lambda_{v} - \lambda_{v+w}) v = (\lambda_{v+w} - \lambda_{w}) w$$ for all $v, w \in V$. It follows that if $v$ and $w$ are linearly independent, then $\lambda_{v+w} = \lambda_{v} = \lambda_{w}$. These two observations together imply $\lambda_{v} = \lambda_{w}= :\lambda$ for all pairs $v, w \in V$, hence $T = \lambda I$.

Answer 10

It is interesting to change this to a question about linear maps on a vector space of matrices so I change notation somewhat
$W:= M_n\big(\mathbb F\big)$ and we have some $A\in W$ such that for all $B\in W$, $AB=BA$.

Now define $T:W\longrightarrow W$ given by
$T\big(B\big) = AB-BA=\mathbf 0\implies \dim \ker T = n^2$

consider $\mathbb F\subseteq \mathbb K$
(where $\mathbb K$ is the algebraic closure of $\mathbb F$ or a splitting field for $A$'s characteristic polynomial depending on preference)

$W':= M_n\big(\mathbb K\big)$
$T':W'\longrightarrow W'$ given by
$T'\big(C\big) = AC-CA$
$\implies \dim \ker T' = n^2$, since $T'$ kills every standard basis vector$\implies T'=\mathbf 0$
$\implies A$ has only one distinct eigenvalue, $\lambda$.
justification: (i) use Kronecker Sum or (ii) argue if $A$ had more than one distinct eigenvalue we may select $C:=\mathbf x\mathbf y^T$ where these are left and right eigenvectors associated with $\lambda_1, \lambda_2$ respectively and $T(C)=(\lambda_1-\lambda_2)\cdot C\neq \mathbf 0$.

Thus $A- \lambda I = N$. Being nilpotent $N$ is similar to both a strictly upper triangular matrix and a strictly lower triangular matrix via some $S, S'\in GL_n(\mathbb K)$ respectively but $T'=\mathbf 0$ implies $A$ commutes with $S$ and $S'$ hence $N$ is both strictly upper and lower triangular, i.e. $N=\mathbf 0$ and $A= \lambda I$ for some $\lambda \in \mathbb F$.