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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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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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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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1992
Unless otherwise noted, we will assume throughout this chapter that all vector spaces are finite dimensional.
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Unless otherwise noted, we will assume throughout this chapter that all vector spaces are finite dimensional.
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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 \({\mathcal{L}}_{1}\).
Hasan Ersel +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 \({\mathcal{L}}_{1}\).
Hasan Ersel +2 more
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1996
As mentioned in the previous chapter, besides the significance of matrix simplification through elimination, it is often important to simplify a matrix by preserving its eigenvalues. Eigenvalues and their associated eigenvectors are useful in a variety of situations.
Anastasios A. Tsonis, James B. Elsner
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As mentioned in the previous chapter, besides the significance of matrix simplification through elimination, it is often important to simplify a matrix by preserving its eigenvalues. Eigenvalues and their associated eigenvectors are useful in a variety of situations.
Anastasios A. Tsonis, James B. Elsner
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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 ...
openaire +2 more sources