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2017
This chapter begins with the basic theory of eigenvalues and eigenvectors of matrices. Essential concepts such as characteristic polynomials, the Fundamental Theorem of Algebra, the Gerschgorin circle theorem, invariant subspaces, change of basis, spectral radius and the distance between subspaces are developed.
James R. Kirkwood, Bessie H. Kirkwood
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This chapter begins with the basic theory of eigenvalues and eigenvectors of matrices. Essential concepts such as characteristic polynomials, the Fundamental Theorem of Algebra, the Gerschgorin circle theorem, invariant subspaces, change of basis, spectral radius and the distance between subspaces are developed.
James R. Kirkwood, Bessie H. Kirkwood
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1986
Recall that an n × n matrix B is similar to an n × n matrix A if there is an invertible n × n matrix P such that B = P −1 AP. Our objective now is to determine under what conditions an n × n matrix is similar to a diagonal matrix. In so doing we shall draw together all of the notions that have been previously developed.
T. S. Blyth, Edmund F. Robertson
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Recall that an n × n matrix B is similar to an n × n matrix A if there is an invertible n × n matrix P such that B = P −1 AP. Our objective now is to determine under what conditions an n × n matrix is similar to a diagonal matrix. In so doing we shall draw together all of the notions that have been previously developed.
T. S. Blyth, Edmund F. Robertson
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Eigenvectors and Eigenvalues [PDF]
, 1986 This chapter gives the basic elementary properties of eigenvectors and eigenvalues. We get an application of determinants in computing the characteristic polynomial. In §3, we also get an elegant mixture of calculus and linear algebra by relating eigenvectors with the problem of finding the maximum and minimum of a quadratic function on the sphere ...
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1987
In this chapter we describe numerical techniques for the calculation of a scalar λ and non-zero vector x in the equation $$ Ax = \lambda x $$ (4.1) where A is a given n × n matrix. The quantities λ and x are usually referred to as an eigenvalue and an eigenvector of A.
Colin Judd, Ian Jacques
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In this chapter we describe numerical techniques for the calculation of a scalar λ and non-zero vector x in the equation $$ Ax = \lambda x $$ (4.1) where A is a given n × n matrix. The quantities λ and x are usually referred to as an eigenvalue and an eigenvector of A.
Colin Judd, Ian Jacques
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1997
Gaussian elimination plays a fundamental role in solving a system Ax = b of linear equations. In order to solve a system of linear equations, Gaussian elimination reduces the augmented matrix to a (reduced) row-echelon form by using elementary row operations that preserve row and null spaces.
Sungpyo Hong, Jin Ho Kwak
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Gaussian elimination plays a fundamental role in solving a system Ax = b of linear equations. In order to solve a system of linear equations, Gaussian elimination reduces the augmented matrix to a (reduced) row-echelon form by using elementary row operations that preserve row and null spaces.
Sungpyo Hong, Jin Ho Kwak
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2019
In this chapter we present the main results regarding eigenvalues and eigenvectors of compact and/or symmetric operators. This includes the Hilbert–Schmidt Theorem and its applications to the main eigenvalue problems for the Laplacian.
Gheorghe Moroşanu, Gheorghe Moroşanu
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In this chapter we present the main results regarding eigenvalues and eigenvectors of compact and/or symmetric operators. This includes the Hilbert–Schmidt Theorem and its applications to the main eigenvalue problems for the Laplacian.
Gheorghe Moroşanu, Gheorghe Moroşanu
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2012
In this chapter we study another major branch of linear algebra, very different from what we have seen so far. The problems in this area arise in many applications in physics, economics, statistics, and other fields. The main reason for this phenomenon can be explained roughly as follows.
C. Phillips, C. Woodford
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In this chapter we study another major branch of linear algebra, very different from what we have seen so far. The problems in this area arise in many applications in physics, economics, statistics, and other fields. The main reason for this phenomenon can be explained roughly as follows.
C. Phillips, C. Woodford
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Eigenvectors and Eigenvalues [PDF]
, 1995 We are still in the midst of considering the following problem: given a vector space V finitely generated over a field F and given an endomorphism α of V, we want to find a basis for V relative to which α can be represented in a “nice” manner. In Chapter 10 we saw that if V has a basis composed of eigenvectors of α then, relative to that basis, α is ...
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2007
Eigenvalues and the associated eigenvectors of an endomorphism of a vector space are defined and studied, as is the spectrum of an endomorphism. The characteristic polynomial of a matrix is considered and used to define the characteristic polynomial of the endomorphism of a finitely-generated vector space.
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Eigenvalues and the associated eigenvectors of an endomorphism of a vector space are defined and studied, as is the spectrum of an endomorphism. The characteristic polynomial of a matrix is considered and used to define the characteristic polynomial of the endomorphism of a finitely-generated vector space.
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1969
Publisher Summary This chapter focuses on eigenvalues and eigenvectors. In general, it is very difficult to find the eigenvalues of a matrix. First, the characteristic equation must be obtained. For the matrices of high order, this in itself is a lengthy task. Once the characteristic equation is determined, it must be solved for its roots.
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Publisher Summary This chapter focuses on eigenvalues and eigenvectors. In general, it is very difficult to find the eigenvalues of a matrix. First, the characteristic equation must be obtained. For the matrices of high order, this in itself is a lengthy task. Once the characteristic equation is determined, it must be solved for its roots.
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