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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/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/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_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/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 \(U\), \(\Sigma\), 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 \(U\), \(\Sigma\), and \(V\). Two detailed examples are provided, along with a fifth exercise concerning the determinant of an \(n \times n\) matrix and its singular values.
- https://math.libretexts.org/Bookshelves/Linear_Algebra/Fundamentals_of_Matrix_Algebra_(Hartman)/04%3A_Eigenvalues_and_Eigenvectors/4.02%3A_Properties_of_Eigenvalues_and_EigenvectorsIn this section we’ll explore how the eigenvalues and eigenvectors of a matrix relate to other properties of that matrix. This section is essentially a hodgepodge of interesting facts about eigenvalue...In this section we’ll explore how the eigenvalues and eigenvectors of a matrix relate to other properties of that matrix. This section is essentially a hodgepodge of interesting facts about eigenvalues; the goal here is not to memorize various facts about matrix algebra, but to again be amazed at the many connections between mathematical concepts.
- https://math.libretexts.org/Bookshelves/Linear_Algebra/Interactive_Linear_Algebra_(Margalit_and_Rabinoff)/05%3A_Eigenvalues_and_Eigenvectors/5.3%3A_SimilarityThis page explores similar matrices defined by the relation \(A = CBC^{-1}\), focusing on their geometric interpretations, eigenvalues, eigenvectors, and properties as an equivalence relation. It expl...This page explores similar matrices defined by the relation \(A = CBC^{-1}\), focusing on their geometric interpretations, eigenvalues, eigenvectors, and properties as an equivalence relation. It explains how to compute matrix powers, emphasizing transformations and changes of coordinates between different systems. The relationship between matrices \(A\) and \(B\) is examined, highlighting how they share characteristics like trace and determinant but may differ in eigenvectors.
- 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/Courses/Cosumnes_River_College/Math_420%3A_Differential_Equations_(Breitenbach)/11%3A_Appendices/06%3A_Eigenvalues_and_Eigenvectors\[\begin{align}\begin{aligned}\left[\begin{array}{cc}{1}&{4}\\{2}&{3}\end{array}\right]\left[\begin{array}{c}{2}\\{2}\end{array}\right]&=\left[\begin{array}{c}{10}\\{10}\end{array}\right]=5\left[\begi...\[\begin{align}\begin{aligned}\left[\begin{array}{cc}{1}&{4}\\{2}&{3}\end{array}\right]\left[\begin{array}{c}{2}\\{2}\end{array}\right]&=\left[\begin{array}{c}{10}\\{10}\end{array}\right]=5\left[\begin{array}{c}{2}\\{2}\end{array}\right]; \\ \left[\begin{array}{cc}{1}&{4}\\{2}&{3}\end{array}\right]\left[\begin{array}{c}{7}\\{7}\end{array}\right]&=\left[\begin{array}{c}{35}\\{35}\end{array}\right]=5\left[\begin{array}{c}{7}\\{7}\end{array}\right]; \\ \left[\begin{array}{cc}{1}&{4}\\{2}&{3}\end{a…
