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5.5: Similarity and Diagonalization
https://math.libretexts.org/Bookshelves/Linear_Algebra/Linear_Algebra_with_Applications_(Nicholson)/05%3A_Vector_Space_R/5.05%3A_Similarity_and_Diagonalization
Hence the eigenvalues are $\lambda_{1} = i$ and $\lambda_{2} = -i$ , with corresponding eigenvectors $\mathbf{x}_1 = \left[ \begin{array}{r} 1 \\ -i \end{array} \right]$ and \(\mathbf{x}_2 = \lef...Hence the eigenvalues are $\lambda_{1} = i$ and $\lambda_{2} = -i$ , with corresponding eigenvectors $\mathbf{x}_1 = \left[ \begin{array}{r} 1 \\ -i \end{array} \right]$ and $\mathbf{x}_2 = \left[ \begin{array}{r} 1 \\ i \end{array} \right].$ Hence $A$ is diagonalizable by the complex version of Theorem [thm:016145], and the complex version of Theorem [thm:016068] shows that \(P = \left[ $\begin{array}{cc} \mathbf{x}_1\ & \mathbf{x}_2 \end{array}$ \right]= \left[ \begin{array}{rr} 1 & 1 …
6.4E: Exercises for Section 6.4
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_6.4
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 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.
6.4: Applications of Spectral Theory
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
This section considers some applications of matrix diagonalization.
6.7: Orthogonal Diagonalization
https://math.libretexts.org/Courses/De_Anza_College/Linear_Algebra%3A_A_First_Course/06%3A_Spectral_Theory/6.07%3A_Orthogonal_Diagonalization
In this section we look at matrices that have an orthonormal set of eigenvectors.
6: Spectral Theory
https://math.libretexts.org/Courses/De_Anza_College/Linear_Algebra%3A_A_First_Course/06%3A_Spectral_Theory
This 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.
7.3: Diagonal matrices
https://math.libretexts.org/Bookshelves/Linear_Algebra/Book%3A_Linear_Algebra_(Schilling_Nachtergaele_and_Lankham)/07%3A_Eigenvalues_and_Eigenvectors/7.03%3A_Diagonal_matrices
Note that if $T$ has $n=\dim(V)$ distinct eigenvalues, then there exists a basis $(v_1,\ldots,v_n)$ of $V$ such that \begin{equation*} Tv_j = \lambda_j v_j, \quad \text{for all $j=1,2,\ldots,n...Note that if $T$ has $n=\dim(V)$ distinct eigenvalues, then there exists a basis $(v_1,\ldots,v_n)$ of $V$ such that $\begin{equation*} Tv_j = \lambda_j v_j, \quad \text{for all \(j=1,2,\ldots,n$.} \end{equation*}$ This means that the matrix $M(T)$ for $T$ with respect to the basis of eigenvectors $(v_1,\ldots,v_n)$ is diagonal, and so we call $T$ diagonalizable: $\begin{equation*} M(T) = \begin{bmatrix} \lambda_1 & & 0 \\ & \ddots & \\ 0& & \lambda_n \end{bmatrix}. \end{equation*}$
3.3: Diagonalization and Eigenvalues
https://math.libretexts.org/Bookshelves/Linear_Algebra/Linear_Algebra_with_Applications_(Nicholson)/03%3A_Determinants_and_Diagonalization/3.03%3A_Diagonalization_and_Eigenvalues
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 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.
5.3: Similarity
https://math.libretexts.org/Bookshelves/Linear_Algebra/Interactive_Linear_Algebra_(Margalit_and_Rabinoff)/05%3A_Eigenvalues_and_Eigenvectors/5.3%3A_Similarity
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 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.
6.3: Diagonalization
https://math.libretexts.org/Courses/Mission_College/MAT_04C_Linear_Algebra_(Kravets)/06%3A_Eigenvalues_and_Eigenvectors/6.03%3A_Diagonalization
Diagonal matrices are the easiest kind of matrices to understand: they just scale the coordinate directions by their diagonal entries. This section is devoted to the question: “When is a matrix simila...Diagonal matrices are the easiest kind of matrices to understand: they just scale the coordinate directions by their diagonal entries. This section is devoted to the question: “When is a matrix similar to a diagonal matrix?” This section is devoted to the question: “When is a matrix similar to a diagonal matrix?” We will see that the algebra and geometry of such a matrix is relatively easy to understand.
13.1: Diagonalization
https://math.libretexts.org/Bookshelves/Linear_Algebra/Map%3A_Linear_Algebra_(Waldron_Cherney_and_Denton)/13%3A_Diagonalization/13.01%3A_Diagonalization
In a basis of eigenvectors, the matrix of a linear transformation is diagonal
5: Eigenvalues and Eigenvectors
https://math.libretexts.org/Bookshelves/Linear_Algebra/Interactive_Linear_Algebra_(Margalit_and_Rabinoff)/05%3A_Eigenvalues_and_Eigenvectors
This 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.

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