Understanding the Eigenvalue Problem for a 4x4 Matrix with Rank 1 and Trace 10

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A 4x4 matrix A with rank 1 and trace 10 has specific eigenvalue characteristics, primarily that it will have one non-zero eigenvalue and three zero eigenvalues. The non-zero eigenvalue equals the trace, which is 10, indicating that the eigenvalues are 10, 0, 0, and 0. The rank of 1 implies that the matrix can be expressed as an outer product of two vectors, simplifying the process of finding its eigenvalues. Understanding the image and dimension of the matrix is crucial, as rank directly influences eigenvalue multiplicity. The discussion emphasizes the importance of visualizing the matrix structure to facilitate solving for eigenvalues.
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Homework Statement



Let there be a 4X4 Matrix A with dim(im(A), or rank = 1 , and trace=10. What are the Eigenvalues of A? Are there any multiplicities?

The Attempt at a Solution



While I understand that the trace of a matrix that's 4X4 = the sum of the diagonal elements, I'm confused about how to create this matrix A to find the eigenvalues, and how to represent the given fact that dim(im(A)) = 1.

Any tips/pointers would be helpful!
 
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You need to review what is meant by the image and dimension of a matrix: what does rank=1 tell you?
Also - do you know of any special cases where the eigenvalues have some simple relation to the trace?
https://www.physicsforums.com/showthread.php?t=682216

Since it is 4x4 - you can always just assign 16 variables and sketch it out.
Needs lots of paper or a large whiteboard (or window).
 
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