Linear homomorphisms of square matrices are conjugations

Question

I was doing some linear algebra exercises and came across the following tough problem :

Let $M_{n\times n}(\mathbf{R})$ denote the set of all the matrices whose entries are real numbers. Suppose $\phi:M_{n\times n}(\mathbf{R})\to M_{n\times n}(\mathbf{R})$ is a nonzero linear transform (i.e. there is a matrix $A$ such that $\phi(A)\neq 0$) such that for all $A,B\in M_{n\times n}(\mathbf{R})$ $$\phi(AB)=\phi(A)\phi(B).$$ Prove that there exists a invertible matrix $T\in M_{n\times n}(\mathbf{R})$ such that $$\phi(A)=TAT^{-1}$$ for all $A\in M_{n\times n}(\mathbf{R})$.

This is an exercise from my textbook and I am all thumbs when I attempted to solve it .

Can someone tell me as to how should I , at least , start the problem ?

Answer 1

This kind of problems are known as linear preserver problems in the literature. The following is a sketch of proof that immediately comes to my mind. Certainly there are simpler ways to solve the problem (especially if one makes use of existing results on linear preserver problems), but anyway, let $\{e_1,\ldots,e_n\}$ be the standard basis of $\mathbb R^n$ and $E_{ij}=e_ie_j^T$.

1. Prove that $\phi$ is injective. Hint. Suppose the contrary that $\phi(X)=0$ for some matrix $X$ whose $(r,s)$-th entry is nonzero. Now consider $\phi(E_{ir}XE_{sj})$ for every $(i,j)$.

2. Prove that

   • $\phi$ preserves non-invertibility (hint: if $X$ is singular, then $XY=0$ for some nonzero $Y$),
   • $\phi$ preserves invertibility (hint: if $\phi(P)$ is singular for some invertible $P$, then $\phi(P)Y=0$ for some nonzero matrix $Y$; since $\phi$ is an injective linear operator over a finite dimensional vector space, $Y=\phi(B)$ for some nonzero $B$, but then ...),
   • $\phi(I)=I$.

3. This is the only interesting step in the whole proof: show that every $\phi(E_{ii})$ is a rank-1 idempotent matrix. Hint: the rank of an idempotent matrix is equal to its trace.

4. Argue that without loss of generality, we may assume that $\phi(E_{11})=E_{11}$.

5. Show that whenever $i,j\ne1$, the first column and the first row of $\phi(E_{ij})$ are zero (hint: $E_{ij}E_{11}=0=E_{11}E_{ij}$). By mathematical induction/recursion, show that we may further assume that $\phi(E_{ii})=E_{ii}$ for every $i$.

6. For any off-diagonal coordinate pair $(i,j)$, show that $\phi(E_{ij})$ is a scalar multiple of $E_{ij}$ (hint: we have $E_{kk}E_{ij}=0$ for every $k\ne i$ and $E_{ij}E_{kk}=0$ for every $k\ne j$).

7. Hence prove that in addition to all the previous assumptions (i.e. $\phi(E_{ii})=E_{ii}$ and $\phi(E_{ij})$ is a scalar multiple of $E_{ij}$ for every $i,j$), we may further assume that $\phi(E_{\color{red}{1}j})=E_{\color{red}{1}j}$ for every $j$.

8. Since $\phi$ preserves invertibility and non-invertibility, prove that $\phi(E_{ij})=E_{ij}$ for every $(i,j)$.