Why do the eigenvalues of a triangular matrix lie on its diagonal?

Suppose A is a triangular matrix, so A is invertible. Now with eigenvalues finding process, we can only shift each A‘s diagonal entry for a \lambda to make A non-invertible. And the only possible way is to shift each A‘s diagonal entry for a \lambda that is also an A‘s diagonal entry, so after that one diagonal entry of A will equal to zero and A is then non-invertible.

Hence the eigenvalues of a triangular matrix lie on its diagonal.


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