MHB Find Eigenvalues & Basis C2 Matrix: Help!

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The discussion centers on finding eigenvalues for a C2 matrix, with the key point being that the matrix does not have real eigenvalues. Instead, the eigenvalues are complex, specifically $-5 \pm i$. Participants express difficulty in understanding the problem and seek clarification on the solution process. The focus is on grasping the concept of complex eigenvalues in the context of matrix analysis. Understanding these eigenvalues is crucial for solving related mathematical problems.
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Good afternoon to all again! I'm solving last year's problems and can't cope with this problem:( help me to understand the problem and find a solution!
 

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That matrix does NOT have real eigenvalues. The eigenvalues are $-5\pm i$
 

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