Results 291 to 300 of about 36,403 (314)
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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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2017
In the previous chapters, we have defined some numbers associated to a matrix, such as the determinant, trace, and rank. In this chapter, we focus on scalars and vectors known as eigenvalues and eigenvectors. The eigenvalues and eigenvectors have many important applications, in particular, in the study of differential equations.
James R. Kirkwood, Bessie H. Kirkwood
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In the previous chapters, we have defined some numbers associated to a matrix, such as the determinant, trace, and rank. In this chapter, we focus on scalars and vectors known as eigenvalues and eigenvectors. The eigenvalues and eigenvectors have many important applications, in particular, in the study of differential equations.
James R. Kirkwood, Bessie H. Kirkwood
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BIT Numerical Mathematics, 2003
We show under very general assumptions that error bounds for an individual eigenvector of a matrix can be computed if and only if the geometric multiplicity of the corresponding eigenvalue is one. Basically, this is true if not computing exactly like in computer algebra methods. We first show, under general assumptions, that nontrivial error bounds are
Jens-Peter M. Zemke, Siegfried M. Rump
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We show under very general assumptions that error bounds for an individual eigenvector of a matrix can be computed if and only if the geometric multiplicity of the corresponding eigenvalue is one. Basically, this is true if not computing exactly like in computer algebra methods. We first show, under general assumptions, that nontrivial error bounds are
Jens-Peter M. Zemke, Siegfried M. Rump
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, 1986 Abstract : This grant has supported work in several areas. 1) A study of graph eigenvectors shows connections to graph structure in ways that are reminiscent of eigenfunctions of the laplacian operator in two or three dimensions. Methods developed in this study have also led to estimates of the maximum possible value for the kth eigenvalue of a graph ...
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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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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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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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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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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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