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- https://math.libretexts.org/Bookshelves/Analysis/Supplemental_Modules_(Analysis)/Ordinary_Differential_Equations/5%3A_Systems_of_Differential_Equations/5.1%3A_Review_of_Linear_AlgebraIn this discussion, we expect some familiarity with matrices. We will rely heavily on calculators and computers to work out the problems.
- https://math.libretexts.org/Courses/De_Anza_College/Linear_Algebra%3A_A_First_Course/06%3A_Spectral_Theory/6.02%3A_Eigenvalues_and_Eigenvectors_for_Special_MatricesIn this section we consider three kinds of matrices where we can simplify the process of finding eigenvalues and eigenvectors.
- https://math.libretexts.org/Courses/De_Anza_College/Linear_Algebra%3A_A_First_Course/06%3A_Spectral_Theory/6.05%3A_Markov_MatricesIn this section we look at a particular kind of matrix, called a Markov matrix, and consider some its applications.
- https://math.libretexts.org/Courses/De_Anza_College/Linear_Algebra%3A_A_First_Course/06%3A_Spectral_Theory/6.07%3A_Orthogonal_DiagonalizationIn this section we look at matrices that have an orthonormal set of eigenvectors.
- https://math.libretexts.org/Courses/De_Anza_College/Linear_Algebra%3A_A_First_Course/06%3A_Spectral_Theory/6.08%3A_Singular_Value_Decomposition/6.8E%3A_Exercises_for_Section_6.8This page contains exercises on finding the Singular Value Decomposition (SVD) of various matrices, outlining the computation of matrices U, Σ, and V. Two detailed examples are provid...This page contains exercises on finding the Singular Value Decomposition (SVD) of various matrices, outlining the computation of matrices U, Σ, and V. Two detailed examples are provided, along with a fifth exercise concerning the determinant of an n×n matrix and its singular values.
- https://math.libretexts.org/Courses/De_Anza_College/Linear_Algebra%3A_A_First_Course/06%3A_Spectral_TheoryThis page discusses Eigenvalues and Eigenvectors in Spectral Theory, covering special matrices, diagonalization, applications, Markov matrices, dynamical systems, orthogonal diagonalization, singular ...This page discusses Eigenvalues and Eigenvectors in Spectral Theory, covering special matrices, diagonalization, applications, Markov matrices, dynamical systems, orthogonal diagonalization, singular value decomposition, special factorizations, and quadratic forms. It includes exercises for practice to enhance understanding of both theoretical and practical aspects of these concepts.
- https://math.libretexts.org/Courses/De_Anza_College/Linear_Algebra%3A_A_First_Course/06%3A_Spectral_Theory/6.04%3A_Applications_of_Spectral_Theory/6.4E%3A_Exercises_for_Section_6.4This page features exercises on diagonalizing matrices and solving initial value problems for differential equations. It includes tasks such as computing matrix powers after diagonalization and solvin...This page features exercises on diagonalizing matrices and solving initial value problems for differential equations. It includes tasks such as computing matrix powers after diagonalization and solving first-order systems through matrix exponentiation. Each exercise outlines problem setups, provides hints, and sometimes offers detailed solution computations.
- https://math.libretexts.org/Courses/De_Anza_College/Linear_Algebra%3A_A_First_Course/06%3A_Spectral_Theory/6.04%3A_Applications_of_Spectral_TheoryThis section considers some applications of matrix diagonalization.
- https://math.libretexts.org/Bookshelves/Linear_Algebra/Interactive_Linear_Algebra_(Margalit_and_Rabinoff)/05%3A_Eigenvalues_and_Eigenvectors/5.01%3A_Eigenvalues_and_EigenvectorsThis page explains eigenvalues and eigenvectors in linear algebra, detailing their definitions, significance, and processes for finding them. It discusses how eigenvectors result from matrix transform...This page explains eigenvalues and eigenvectors in linear algebra, detailing their definitions, significance, and processes for finding them. It discusses how eigenvectors result from matrix transformations and the linear independence of distinct eigenvectors. The text covers specific examples, including eigenvalue analysis for specific matrices and the conditions for eigenvalues, including zero.
- https://math.libretexts.org/Bookshelves/Linear_Algebra/Interactive_Linear_Algebra_(Margalit_and_Rabinoff)/05%3A_Eigenvalues_and_Eigenvectors/5.03%3A_DiagonalizationThis page covers diagonalizability of matrices, explaining that a matrix is diagonalizable if it can be expressed as A=CDC−1 with D diagonal. It discusses the Diagonalization Theorem, eig...This page covers diagonalizability of matrices, explaining that a matrix is diagonalizable if it can be expressed as A=CDC−1 with D diagonal. It discusses the Diagonalization Theorem, eigenspaces, eigenvalues, and the significance of linear independence among eigenvectors. Multiple diagonal forms can arise, while geometric and algebraic multiplicities influence diagonalizability.
- https://math.libretexts.org/Bookshelves/Linear_Algebra/Linear_Algebra_with_Applications_(Nicholson)/03%3A_Determinants_and_Diagonalization/3.03%3A_Diagonalization_and_EigenvaluesThe world is filled with examples of systems that evolve in time—the weather in a region, the economy of a nation, the diversity of an ecosystem, etc. Describing such systems is difficult in general a...The world is filled with examples of systems that evolve in time—the weather in a region, the economy of a nation, the diversity of an ecosystem, etc. Describing such systems is difficult in general and various methods have been developed in special cases. In this section we describe one such method, called diagonalization, which is one of the most important techniques in linear algebra.
