Hermitian Matrices It is simpler to begin with matrices with complex numbers. Let x = a + ib, where a, b are real numbers, and i = √ −1. Then, x∗ = a− ib is the complex conjugate of x. In the discussion below, all matrices and numbers are complex-valued unless stated otherwise. Let M be an n× n square matrix with complex entries. Then, λ is an eigenvalue of M if there is a non-zero vector ~v su...

Exercise 4. Let λ be an eigenvalue of A and let Eλ(A) = {x ∈ C|Ax = λx} denote the set of all eigenvectors of A associated with λ (including the zero vector, which is not really considered an eigenvector). Show that this set is a (nontrivial) subspace of C. Definition 5. Given A ∈ Cm×m, the function pm(λ) = det(λI − A) is a polynomial of degree at most m. This polynomial is called the character...

We continue with F being an arbitrary field and V a finite dimensional vector space over F ; dimV = n. Conditions on F may be added later on. We assume T ∈ L(V ). Definition 1 We say that an element λ ∈ F is an eigenvalue of T iff there exists x ∈ V , x ̸= 0 such that Tx = λx. We call x ∈ V , x ̸= 0 such that Tx = λx an eigenvector of T , corresponding to the eigenvector λ. If λ is an eigenvector...

Real eigenvectors of real matrix is extended to fuzzy eigenvectors. The structure of fuzzy eigenspaces and relationships between real eigenspaces and fuzzy eigenspaces of real matrix are studied. Using fuzzy eigenvector, we give a sufficient condition to existence of solution of fuzzy linear systems.

Power Method We now describe the power method for computing the dominant eigenpair. Its extension to the inverse power method is practical for finding any eigenvalue provided that a good initial approximation is known. Some schemes for finding eigenvalues use other methods that converge fast, but have limited precision. The inverse power method is then invoked to refine the numerical values and...

Hydroelectric power plants are known for their ability to cover variations of the consumption in electrical power networks. In order to follow this changing demand, hydraulic machines are subject to off-design operation. In that case, the swirling flow leaving the runner of a Francis turbine may act under given conditions as an excitation source for the whole hydraulic system. In high load oper...

For a large class of linear neutral type systems the problem of eigenvalues and eigenvectors assignment is investigated, i.e. finding the system which has the given spectrum and almost all, in some sense, eigenvectors.

Eigenvectors of large matrices (and graphs) play an essential role in combinatorics and theoretical computer science. The goal of this survey is to provide an up-to-date account on properties of eigenvectors when the matrix (or graph) is random.

How can we search for low dimensional structure in high dimensional data? If the data is mainly confined to a low dimensional subspace, then simple linear methods can be used to discover the subspace and estimate its dimensionality. More generally, though, if the data lies on (or near) a low dimensional submanifold, then its structure may be highly nonlinear, and linear methods are bound to fai...

Analysis of nonlinear autonomous systems has been a popular field of study in recent decades. As an interesting nonlinear behavior, chaotic dynamics has been intensively investigated since Lorenz discovered the first physical evidence of chaos in his famous equations. Although many chaotic systems have been ever reported in the literature, a systematic and qualitative approach for chaos generat...