The problem provided is written in Chinese and involves matrix algebra. Here's a breakdown of the problem:

Problem Description:

• $A'A = AA'$, where $A$ is a $2 \times 2$ matrix.
• There exists a diagonal matrix $B$ and an orthogonal matrix $P$ such that $PA = BP$.

Objective:

To understand and analyze the conditions under which the matrices $A, B,$ and $P$ satisfy the given relationships.

Would you like me to:

1. Solve for $A, B,$ and $P$ explicitly under these conditions?
2. Provide general mathematical insights into the relationships between $A, B,$ and $P$?
3. Explain the properties of orthogonal and diagonal matrices relevant to the problem?

Let me know how you'd like to proceed or if you'd like a detailed solution!

Here are some related questions to explore:

1. What does $A'A = AA'$ imply about $A$? Could it be a symmetric matrix?
2. What are the properties of a diagonal matrix $B$ in the context of matrix diagonalization?
3. How does orthogonal transformation $P$ influence the matrices $A$ and $B$?
4. Can $PA = BP$ be interpreted as a similarity transformation?
5. Is there a connection between these properties and eigenvalues/eigenvectors?

Tip:

For matrix problems, always consider eigenvalue decomposition or singular value decomposition to simplify the analysis.