Results 291 to 300 of about 197,057 (337)
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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.
Jin Ho Kwak, Sungpyo Hong
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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.
Jin Ho Kwak, Sungpyo Hong
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2014
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.
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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.
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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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1998
Abstract Rather than giving the formal definition of eigenvalues and eigenvectors — the subject of this chapter, indeed of the rest of the book — straight away, we shall give a hypothetical example of their use to motivate their study.
Richard Kaye, Robert Wilson
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Abstract Rather than giving the formal definition of eigenvalues and eigenvectors — the subject of this chapter, indeed of the rest of the book — straight away, we shall give a hypothetical example of their use to motivate their study.
Richard Kaye, Robert Wilson
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1993
This chapter introduces and, to a limited extent, solves one of the classical problems associated with linear processes: their decomposition into well-behaved, independent component subprocesses. What is especially noteworthy and exciting about the material is that it uses all of the major concepts introduced so far, including the representation of ...
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This chapter introduces and, to a limited extent, solves one of the classical problems associated with linear processes: their decomposition into well-behaved, independent component subprocesses. What is especially noteworthy and exciting about the material is that it uses all of the major concepts introduced so far, including the representation of ...
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2016 IEEE International Conferences on Big Data and Cloud Computing (BDCloud), Social Computing and Networking (SocialCom), Sustainable Computing and Communications (SustainCom) (BDCloud-SocialCom-SustainCom), 2016
Hao Ji +4 more
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Hao Ji +4 more
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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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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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1996
Let a ∈ and a ≠0. Prove that the eigenvectors of the matrix $$ \left( {\begin{array}{*{20}c} 1 & a \\ 0 & 1 \\ \end{array} } \right) $$ generate a 1-dimensional space, and give a basis for this space.
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Let a ∈ and a ≠0. Prove that the eigenvectors of the matrix $$ \left( {\begin{array}{*{20}c} 1 & a \\ 0 & 1 \\ \end{array} } \right) $$ generate a 1-dimensional space, and give a basis for this space.
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Combining eigenvalues and variation of eigenvectors for order determination
, 2016Wei Luo, Bing Li
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