Question

Show that the following conditions are equivalent for a linear transformation $T:{\mathbf{M}}_{22}\to {\mathbf{M}}_{22}$. 1\. $\mathrm{tr}[T(A)]=\mathrm{tr}A$ for all $A$ in ${\mathbf{M}}_{22}$. $$\text{ 2. }T\left[\begin{array}{ll}{r}_{11}& r_{12}\\ r_{21}& r_{22}\end{array}\right]=r_{11}{B}_{11}+r_{12}{B}_{12}+r_{21}{B}_{21}+$$ $r_{22}{B}_{22}$ for matrices $B_{ij}$ such that $\mathrm{tr}{B}_{11}=1=\mathrm{tr}{B}_{22}$ and $\mathrm{tr}{B}_{12}=0=\mathrm{tr}{B}_{21}$

Short answer

The conditions are equivalent because both stipulate a trace-preserving linear transformation characterized by specific trace properties of the transform matrices.

Step-by-step solution

Understand Definitions

To show the equivalence of two conditions, we need to understand the definitions involved. We have a linear transformation $T:{\mathbf{M}}_{22}\to {\mathbf{M}}_{22}$, where ${\mathbf{M}}_{22}$ denotes the set of all $2\times 2$ matrices. The trace function $\mathrm{tr}$ of a matrix is the sum of its diagonal elements. We will show the conditions that $\mathrm{tr}[T(A)]=\mathrm{tr}A$ and $T$ being expressible with specific matrices $B_{ij}$ are equivalent.

Show Condition 1 Implies Condition 2

Assume Condition 1: $\mathrm{tr}[T(A)]=\mathrm{tr}A$. Each matrix $A$ in ${\mathbf{M}}_{22}$ can be expressed in terms of its elements as $A=\left(\begin{array}{ccc}{r}_{11}& r_{12}{r}_{21}& r_{22}\end{array}\right)$. The trace property in Condition 1 implies that $T$ maps a matrix $A$ while preserving the sum of its diagonal elements. Thus, we propose that $T(A)=r_{11}{B}_{11}+r_{12}{B}_{12}+r_{21}{B}_{21}+r_{22}{B}_{22}$, where the trace of $B_{11}$ and $B_{22}$ must be 1, and $B_{12}$ and $B_{21}$ must have trace 0, fulfilling the trace-preserving condition.

Verify Trace Conditions for B Matrices

For Condition 2 to imply $\mathrm{tr}[T(A)]=\mathrm{tr}A$, the matrices $B_{ij}$ must satisfy the trace properties: $\mathrm{tr}{B}_{11}=1$, $\mathrm{tr}{B}_{22}=1$, $\mathrm{tr}{B}_{12}=0$, $\mathrm{tr}{B}_{21}=0$. Examining the expression $T(A)=r_{11}{B}_{11}+r_{12}{B}_{12}+r_{21}{B}_{21}+r_{22}{B}_{22}$, we find that the trace operation gives $\mathrm{tr}[T(A)]=r_{11}\mathrm{tr}(B_{11})+r_{22}\mathrm{tr}(B_{22})=r_{11}+r_{22}=\mathrm{tr}A$, confirming that if these $B$ matrices exist, $T$ must be defined as such to preserve trace.

Conclusion of Equivalence

With the above demonstration, we've shown that Condition 1 directly leads to a matrix expression for $T$ as laid out in Condition 2, and vice versa, via the trace-preserving property of $B_{ij}$ matrices. Thus, the two conditions are indeed equivalent.

Key concepts

Matrix Trace

The trace of a matrix is a straightforward yet significant concept in linear algebra. It is defined as the sum of the elements along the main diagonal of a square matrix. For a matrix $A$, the trace is denoted as $\mathrm{tr}(A)$. In mathematical terms, if $A=\left(\begin{array}{ccc}a& bc& d\end{array}\right)$, then $\mathrm{tr}(A)=a+d$.
Understanding matrix trace is pivotal, especially when dealing with transformations. It acts as a preservation tool, implying that certain operations do not alter the sum of the diagonal elements. This property is crucial in proving the equivalence of conditions in transformations. In our context, the trace condition $\mathrm{tr}[T(A)]=\mathrm{tr}A$ ensures that a transformation $T$ maintains the trace whenever it is applied to a matrix $A$. This preservation indicates a deep connection between a linear transformation and the intrinsic properties of matrices involved.

2x2 Matrices

The concept of a $2\times 2$ matrix involves a simple grid of numbers consisting of two rows and two columns. These matrices are a basic yet powerful tool in linear algebra. An example of a $2\times 2$ matrix is $A=\left(\begin{array}{ccc}{r}_{11}& r_{12}{r}_{21}& r_{22}\end{array}\right)$. Here, $r_{11},r_{12},r_{21},$ and $r_{22}$ are elements of the matrix.
In the context of linear transformations, $2\times 2$ matrices are often manipulated to understand properties like trace preservation. By expressing matrices in terms of their elements, we can conveniently apply rules and transformations. For example, when examining linear transformation $T(A)$, each element$r_{ij}$ is considered separately, allowing the transformation to hold specific conditions like maintaining the trace sum. This structural understanding aids in exploring the relationships between conditions set forth in matrix transformations.

Equivalent Conditions

In mathematics, proving the equivalence of two conditions often involves demonstrating that each condition can be derived or inferred from the other. For the concept of linear transformations, this equivalence can be crucial in understanding how a transformation affects a matrix.
The given problem states two conditions deemed equivalent under a linear transformation $T:{\mathbf{M}}_{22}\to {\mathbf{M}}_{22}$. The first condition requires that the trace of the transformed matrix equals the trace of the original matrix, $\mathrm{tr}[T(A)]=\mathrm{tr}(A)$. The second condition presents a particular way of expressing the transformation with matrices $B_{ij}$, each having specified trace conditions.
To establish equivalence, one needs to show that assuming one condition logically leads to the other. In the proof, this involves using the trace and given expressions of $T(A)$ to maintain the trace property. The concept emphasizes that two differently stated criteria can ultimately represent the same underlying property in mathematical transformations.