Question

If $T:{\mathbf{M}}_{nn}\to \mathbb{R}$ is any linear transformation satisfying $T(AB)=T(BA)$ for all $A$ and $B$ in ${\mathbf{M}}_{nn}$ show that there exists a number $k$ such that $T(A)=k\mathrm{tr}A$ for all A. (See Lemma 5.5.1.) [Hint: Let $E_{ij}$ denote the $n\times n$ matrix with 1 in the $(i,j)$ position and zeros elsewhere. Show that \(E_{i k} E_{l j}=\left\{$$\begin{array}{cl}0& \text{ if }k\ne l\\ E_{ij}& \text{ if }k=l\end{array}$$\right.\). Use this to show that $T\left(E_{ij}\right)=0$ if $i\ne j$ and $T\left(E_{11}\right)=T\left(E_{22}\right)=\cdots =T\left(E_{nn}\right).$ Put $k=T\left(E_{11}\right)$ and use the fact that $\text{Missing left or extra right}$ is a basis of ${\mathbf{M}}_{nn}$.

Short answer

There exists a number $k$ such that for any matrix $A$, $T(A)=k\mathrm{tr}A$.

Step-by-step solution

Define the Basis Matrices

Consider matrices $E_{ij}$, which are $n\times n$ matrices with 1 in the $(i,j)$ position and 0 elsewhere. These form a basis of ${\mathbf{M}}_{nn}$. Each matrix $E_{ij}$ can be used to construct any matrix in ${\mathbf{M}}_{nn}$.

Examine Product of Basis Matrices

Calculate the product $E_{ik}{E}_{lj}$. Observing the entries, $E_{ik}{E}_{lj}=0$ if $keql$ and $E_{ik}{E}_{lj}=E_{ij}$ if $k=l$. This property will be useful to apply the transformation $T$.

Apply Linear Transformation Property

Use the property $T(AB)=T(BA)$ and apply it to basis matrices. For $ieqj$, consider $A=E_{ij}$ and $B=E_{ji}$: $T(E_{ij}{E}_{ji})=T(E_{ji}{E}_{ij})=T(E_{ii})$. But $E_{ij}{E}_{ji}=0$ (as $ieqj$), hence $T(0)=T(E_{ii})=T(E_{jj})$.

Consider Diagonal Matrices

Applying this to diagonal matrices $E_{ii}$, observe that $T(E_{ii})=T(E_{jj})$ for all $i,j$. Let $k=T(E_{11})=T(E_{22})=\dots =T(E_{nn})$.

Express a General Matrix and Apply the Transformation

Any matrix $A\in {\mathbf{M}}_{nn}$ can be expressed as $\sum a_{ij}{E}_{ij}$, where $a_{ij}$ are its entries. Using the linearity of $T$, $T(A)=\sum a_{ij}T(E_{ij})$. For $ieqj$, $T(E_{ij})=0$, hence $T(A)=\sum _i{a}_{ii}k$.

Relate to Trace

Recall that $\text{tr}(A)=\sum _i{a}_{ii}$. Hence, $T(A)=k\cdot \text{tr}(A)$ for all matrices $A$ in ${\mathbf{M}}_{nn}$. This completes the proof that such a linear transformation can be expressed in terms of the trace of the matrix.

Key concepts

Matrix Basis

In the context of matrices, a **matrix basis** is a set of matrices such that any matrix can be expressed as a linear combination of the basis matrices. For the space of all $n\times n$ matrices, denoted by ${\mathbf{M}}_{nn}$, the standard basis consists of matrices $E_{ij}$, where each matrix has a 1 in the $(i,j)$ position and 0s elsewhere.
These basis matrices play a crucial role in representing any matrix as a combination of simpler parts. It is similar to how vectors in vector spaces are expressed as linear combinations of basis vectors. By using $E_{ij}$, any matrix $A$ can be rewritten in a sum $A=\sum a_{ij}{E}_{ij}$, where coefficients $a_{ij}$ are the elements of $A$.
This simplification is handy when analyzing linear transformations, as transformations can be applied directly to the basis matrices, and by the property of linearity, extended to any combination of these matrices.

Trace of a Matrix

The **trace of a matrix** is a straightforward concept. It is the sum of the diagonal elements of a square matrix. Mathematically, for a matrix $A$ with entries $a_{ij}$, the trace, denoted $\mathrm{tr}(A)$, is calculated as $\sum _{i=1}^n{a}_{ii}$.
This scalar value has powerful properties in matrix algebra. For instance, the trace is invariant under cyclic permutations, meaning that $\mathrm{tr}(AB)=\mathrm{tr}(BA)$ for any two matrices $A$ and $B$ of appropriate size. This property is particularly useful when dealing with linear transformations that depend on the product of two matrices, as it helps simplify expressions and prove equivalencies, as seen in the exercise but with an emphasis on $T(AB)=T(BA)$.
In studying transformations like $T$ in the exercise, recognizing that transformations can be expressed in terms of the trace helps identify characteristics of the transformation being defined, since the trace provides a uniform descriptor for diagonal elements.

Matrix Multiplication

**Matrix multiplication** is an essential operation where each element of the resulting matrix is a sum of products. If you have two matrices, $A$ and $B$, their product $C=AB$ is found by calculating each element $c_{ij}$ as $c_{ij}=\sum _k{a}_{ik}{b}_{kj}$. This requires that matrices $A$ and $B$ have compatible dimensions. Generally, if $A$ is an $m\times n$ matrix and $B$ is an $n\times p$ matrix, their product $C$ will be an $m\times p$ matrix.
Matrix multiplication is not commutative in general; meaning $ABeqBA$ typically holds. However, in special conditions, such as cyclic permutations that result in same-size matrices, the properties can sometimes align, like in the exercise's characteristic $T(AB)=T(BA)$.
This operation's non-commutative nature means careful attention is required when dealing with transformations, as even slight aberrations in order can significantly change the outcome. Understanding and manipulating correct orderings are vital in achieving desired results in problems involving transformations and basis matrices.