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Propositions from Eigenvalues and eigenvectors

All propositions from Eigenvalues and eigenvectors.

[PROPOSITION]

For a triangular matrix, the eigenvalues are its diagonal entries, counted with their multiplicities.

The matrix $A-\lambda I$ is still triangular, so its determinant is the product of its diagonal entries and no full expansion is needed.

This page builds on Linear transformations (Linear transformations), Kernel, image and rank (Linear transformations) and Matrix representations (Linear transformations).