On Differentiating Eigenvalues and Eigenvectors
نویسندگان
چکیده
منابع مشابه
Notes on Eigenvalues and Eigenvectors
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...
متن کاملOn eigenvalues and eigenvectors of subdirect sums
Some new properties of the eigenvalues of the subdirect sums are presented for the particular case of 1-subdirect sums. In particular, it is shown that if an eigenvalue λ is associated with certain blocks of matrix A or matrix B then λ is also an eigenvalue associated with the 1-subdirect sum A ⊕1 B. Some results concerning eigenvectors of the k-subdirect sum A⊕k B for an arbitrary positive int...
متن کامل4: Eigenvalues, Eigenvectors, Diagonalization
Lemma 1.1. Let V be a finite-dimensional vector space over a field F. Let β, β′ be two bases for V . Let T : V → V be a linear transformation. Define Q := [IV ] ′ β . Then [T ] β β and [T ] ′ β′ satisfy the following relation [T ] ′ β′ = Q[T ] β βQ −1. Theorem 1.2. Let A be an n× n matrix. Then A is invertible if and only if det(A) 6= 0. Exercise 1.3. Let A be an n×n matrix with entries Aij, i,...
متن کاملLecture 8 : Eigenvalues and Eigenvectors
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...
متن کاملEigenvalues, Eigenvectors, and Sundry Decompositions
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...
متن کاملذخیره در منابع من
با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید
ژورنال
عنوان ژورنال: Econometric Theory
سال: 1985
ISSN: 0266-4666,1469-4360
DOI: 10.1017/s0266466600011129