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Matrices and Linear Algebra
Published in William S. Levine, Control System Fundamentals, 2019
For a real symmetric matrix Q, all eigenvalues are real and the corresponding eigenvectors may be chosen with real components. For this case, if λ1 and λ2 are two eigenvalues with λ1 ≠ λ2, the corresponding real eigenvectors u (λ1) ∈ Wλ1, and u (λ2) ∈ Wλ2 are not only linearly independent, they are also orthogonal, (u (Ai), u (A2)) = 0. Further, each maximal eigenspace has dimension equal to the algebraic multiplicity of the associated eigenvalue as a zero of the characteristic polynomial, and each maximal eigenspace has an orthogonal basis of eigenvectors. Thus, for any real symmetric matrix Q, there is an orthogonal basis for ℝn consisting of eigenvectors; by scaling the lengths of the basis vectors to one, an orthonormal basis of eigenvectors is obtained. Thus, Λ = OTQO, where O is an orthogonal matrix. (This may be generalized. If A is a complex Hermitian matrix, i.e., AH = A where AH denotes the combination of conjugation and transposition: AH = (A*)T. Then A has real eigenvalues and there is a basis of ℂn comprised of normalized eigenvectors so that Λ = UHA U, where U is a unitary matrix.)
Matrices and Linear Algebra
Published in William S. Levine, The Control Handbook: Control System Fundamentals, 2017
For a real symmetric matrix Q, all eigenvalues are real and the corresponding eigenvectors may be chosen with real components. For this case, if λ1 and λ2 are two eigenvalues with λ1 ≠ λ2, the corresponding real eigenvectors u(λ1) ∈ Wλ1 and u(λ2) ∈ Wλ2 are not only linearly independent, they are also orthogonal, 〈u(λ1), u(λ2)〉 = 0. Further, each maximal eigenspace has dimension equal to the algebraic multiplicity of the associated eigenvalue as a zero of the characteristic polynomial, and each maximal eigenspace has an orthogonal basis of eigenvectors. Thus, for any real symmetric matrix Q, there is an orthogonal basis for Rm consisting of eigenvectors; by scaling the lengths of the basis vectors to one, an orthonormal basis of eigenvectors is obtained. Thus Λ = OT QO, where O is an orthogonal matrix. (This may be generalized. If A is a complex Hermitian matrix, that is, AH = A, where AH, denotes the combination of conjugation and transposition: AH = (A*)T. Then A has real eigenvalues and there is a basis of ℂn comprised of normalized eigenvectors so that Λ = UHAU, where U is a unitary matrix.)
Artificial neural networks
Published in A. W. Jayawardena, Environmental and Hydrological Systems Modelling, 2013
Note that CX is the covariance matrix of X, which is symmetric and square (M ×M) and whose diagonal terms give the variances of particular measurements. The off-diagonal terms give the covariance between measurement types. Any symmetric matrix can be diagonalized by an orthogonal matrix of its eigenvectors. By selecting P where each row pi is an eigenvector of CX, the following relationship can be written: () CX=PDPT
The mise en scéne of memristive networks: effective memory, dynamics and learning
Published in International Journal of Parallel, Emergent and Distributed Systems, 2018
First we work out a simple exercise which will turn out to be useful later. Specifically, this will be in the case , for which . One key element of the proof which follows below is the analysis of matrix similarity, for which a matrix has similar eigenvalues to another matrix. For instance, although is not a symmetric matrix, it has always real eigenvalues. In order to see this, we note that the eigenvalues of any matrix product has the same eigenvalues of the matrix . In this case, since W is diagonal and positive, the square root of the matrix is simply the square root of the diagonal elements. This is due to the fact that any matrix , for any invertible matrix Q, has the same eigenvalues as those of M. In the following, we use the symbol for similarity, i.e. matrices with similar eigenvalues (). If is symmetric and real, then has real eigenvalues as it is a symmetric matrix. This implies that also has real eigenvalues. In fact, we have that
Decision-Oriented Two-Parameter Fisher Information Sensitivity Using Symplectic Decomposition
Published in Technometrics, 2023
From elementary linear algebra, we know that a real symmetric matrix can be diagonalized by orthogonal matrices: where is the orthogonal eigenvector matrix, that is, , and contains the real eigenvalues. And the solution to (5) can be solved using the standard eigenvalue equation:
Constrained optimal consensus in multi-agent systems with single- and double-integrator dynamics
Published in International Journal of Control, 2020
Amir Adibzadeh, Amir A. Suratgar, Mohammad B. Menhaj, Mohsen Zamani
Let A be an real symmetric matrix with eigenvalues and corresponding eigenvectors . Let denote the span of and denote the orthogonal complement of . Then, .