Results 101 to 110 of about 17,115 (153)
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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.
Ravi P. Agarwal, Cristina Flaut
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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.
Ravi P. Agarwal, Cristina Flaut
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2010
Given a square matrix \( {\rm A} \in \mathbb{C}^{{n \times n}} \), the eigenvalue problem consists in finding a scalar λ (real or complex) and a nonnull vector x such that $${\rm Ax} = \lambda{\rm x}$$ (6.1) Any such λ is called an eigenvalue of A, while x is the associated eigenvector.
Alfio Quarteroni +2 more
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Given a square matrix \( {\rm A} \in \mathbb{C}^{{n \times n}} \), the eigenvalue problem consists in finding a scalar λ (real or complex) and a nonnull vector x such that $${\rm Ax} = \lambda{\rm x}$$ (6.1) Any such λ is called an eigenvalue of A, while x is the associated eigenvector.
Alfio Quarteroni +2 more
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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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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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2011
Let \(\mathcal{L}\) be a linear space, \({\mathcal{L}}_{1}\) be a linear subspace of \(\mathcal{L}\) and A be a linear operator in \(\mathcal{L}\). In general, for any vector \(\mathbf{x} \in {\mathcal{L}}_{1}\), A x may not belong to \({\mathcal{L}}_{1}\).
Fuad Aleskerov +2 more
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Let \(\mathcal{L}\) be a linear space, \({\mathcal{L}}_{1}\) be a linear subspace of \(\mathcal{L}\) and A be a linear operator in \(\mathcal{L}\). In general, for any vector \(\mathbf{x} \in {\mathcal{L}}_{1}\), A x may not belong to \({\mathcal{L}}_{1}\).
Fuad Aleskerov +2 more
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