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- https://math.libretexts.org/Courses/Irvine_Valley_College/Math_26%3A_Introduction_to_Linear_Algebra/03%3A_Eigenvalues_and_Eigenvectors/3.03%3A_Geometry_of_EigenvaluesAn n×n matrix whose characteristic polynomial has n distinct real roots is diagonalizable: it is similar to a diagonal matrix, which is much simpler to analyze. The other possibility is that a mat...An n×n matrix whose characteristic polynomial has n distinct real roots is diagonalizable: it is similar to a diagonal matrix, which is much simpler to analyze. The other possibility is that a matrix has complex roots, and that is the focus of this section. It turns out that such a matrix is similar (in the 2×2 case) to a rotation-scaling matrix, which is also relatively easy to understand.
- https://math.libretexts.org/Courses/De_Anza_College/Linear_Algebra%3A_A_First_Course/06%3A_Spectral_Theory/6.03%3A_Diagonalization/6.3E%3A_Exercises_for_Section_6.3This page presents exercises on finding eigenvalues and eigenvectors for matrices, assessing diagonalizability, and applying the Cayley-Hamilton theorem. Each exercise includes matrices, known eigenva...This page presents exercises on finding eigenvalues and eigenvectors for matrices, assessing diagonalizability, and applying the Cayley-Hamilton theorem. Each exercise includes matrices, known eigenvalues, eigenvectors, and diagonalizability status, along with hints for dealing with complex eigenvalues and deriving the characteristic polynomial. The objective is to help students understand diagonalization and matrix theory through guided challenges and proofs.
- https://math.libretexts.org/Bookshelves/Linear_Algebra/Interactive_Linear_Algebra_(Margalit_and_Rabinoff)/05%3A_Eigenvalues_and_EigenvectorsThis page explores eigenvalues and eigenvectors in linear algebra, detailing their definitions, computations, and applications. Key topics include the characteristic polynomial, diagonalization, compl...This page explores eigenvalues and eigenvectors in linear algebra, detailing their definitions, computations, and applications. Key topics include the characteristic polynomial, diagonalization, complex eigenvalues, and stochastic matrices. The example of rabbit population dynamics demonstrates their long-term system behavior, and the chapter highlights practical applications, such as Google's PageRank algorithm, aimed at fostering a deep understanding of these mathematical concepts.
- https://math.libretexts.org/Courses/De_Anza_College/Linear_Algebra%3A_A_First_Course/06%3A_Spectral_Theory/6.01%3A_Eigenvalues_and_Eigenvectors_of_a_Matrix/6.1E%3A_Exercises_for_Section_6.1This page presents a collection of exercises on eigenvalues and eigenvectors, focusing on their properties under various matrix operations, including powers and scalar multiplication. It explores prac...This page presents a collection of exercises on eigenvalues and eigenvectors, focusing on their properties under various matrix operations, including powers and scalar multiplication. It explores practical calculations for specific 3×3 matrices and theoretical discussions on transformations such as rotations, reflections, and projections. Key findings include conditions for eigenvalue existence and behaviors of complex eigenvalues.
- https://math.libretexts.org/Courses/Mission_College/MAT_04C_Linear_Algebra_(Kravets)/06%3A_Eigenvalues_and_Eigenvectors/6.04%3A_Complex_EigenvaluesAn n×n matrix whose characteristic polynomial has n distinct real roots is diagonalizable: it is similar to a diagonal matrix, which is much simpler to analyze. The other possibility is that a mat...An n×n matrix whose characteristic polynomial has n distinct real roots is diagonalizable: it is similar to a diagonal matrix, which is much simpler to analyze. The other possibility is that a matrix has complex roots, and that is the focus of this section. It turns out that such a matrix is similar (in the 2×2 case) to a rotation-scaling matrix, which is also relatively easy to understand.