Which property of the characteristic polynomial ensures that a square matrix $A$ and its transpose $A^T$ have the same eigenvalues?
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Determinant invariance under transposition
Since $\det(A - \lambda I) = \det((A - \lambda I)^T) = \det(A^T - \lambda I)$, the characteristic polynomials are identical. However, the eigenvectors of $A$ and $A^T$ are generally different.
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- What property of the determinant explains why a square matrix and its transpose share the same characteristic polynomial and eigenvalues?