4: Eigenvalues, Eigenvectors, Diagonalization

نویسنده

  • STEVEN HEILMAN
چکیده

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, j ∈ {1, . . . , n}, and let Sn denote the set of all permutations on n elements. For σ ∈ Sn, let sign(σ) := (−1) , where σ can be written as a composition of N transpositions. Then

برای دانلود رایگان متن کامل این مقاله و بیش از 32 میلیون مقاله دیگر ابتدا ثبت نام کنید

ثبت نام

اگر عضو سایت هستید لطفا وارد حساب کاربری خود شوید

منابع مشابه

Tutorial on Quasi-sparse Eigenvector Diagonalization

Quasi-sparse eigenvector (QSE) diagonalization is a new computational method which finds the low-lying eigenvalues and eigenvectors for a general quantum field Hamiltonian . It is able to handle the exponential increase in the size of Fock space for large systems by exploiting the sparsity of the Hamiltonian. QSE diagonalization can even be applied directly to infinitedimensional systems. The m...

متن کامل

Quasi-sparse eigenvector diagonalization and stochastic error correction

Quasi-sparse eigenvector (QSE) diagonalization is a new computational method which finds approximate low-lying eigenvalues and eigenvectors for a general quantum field Hamiltonian H [1]. It handles the exponential increase in the dimension of Fock space by exploiting the sparsity of the Hamiltonian. The method is most effective when the splitting between low-lying eigenvalues is not too small c...

متن کامل

A Method for Fast Diagonalization of a 2x2 or 3x3 Real Symmetric Matrix

A method is presented for fast diagonalization of a 2x2 or 3x3 real symmetric matrix, that is determination of its eigenvalues and eigenvectors. The Euler angles of the eigenvectors are computed. A small computer algebra program is used to compute some of the identities, and a C++ program for testing the formulas has been uploaded to arXiv.

متن کامل

Electronic Structure Calculations in Plane-wave Codes without Diagonalization

We present an algorithm to reduce the computational complexity for plane-wave codes used in electronic structure calculations. Our proposed algorithm avoids the diagonalization of large Hermitian matrices arising in such problems. The computational time for the diagonalization procedure typically grows as the cube of the number of atoms, or the number of eigenvalues required. To reduce this com...

متن کامل

Electronic Structure Calculations for Plane-wave Codes without Diagonalization

We present an algorithm to reduce the computational complexity for plane-wave codes used in electronic structure calculations. The proposed algorithm avoids the diagonalization of large Hermitian matrices arising in such problems. The computational time for the diagonalization procedure typically grows as the cube of the number of atoms, or the number of eigenvalues required. To reduce this com...

متن کامل

Diagonalization in Parallel Space

Matrix diagonalization is an important component of many aspects of computational science. There are a variety of algorithms to accomplish this task. Jacobi’s algorithm is a good choice for parallel environments. Jacobi’s algorithm consists of a series of matrix plane rotations, the ordering of which can dramatically affect performance. We show a new ordering which cuts the number of necessary ...

متن کامل

ذخیره در منابع من


  با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید

عنوان ژورنال:

دوره   شماره 

صفحات  -

تاریخ انتشار 2014