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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.
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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.
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1971
Publisher Summary This chapter explores a particular invariant property of some mappings. Eigen is a German word meaning special, and an eigenvector of a mapping is a special vector whose direction is invariant under, or reversed by, the mapping. The chapter discusses the mappings of the x–y-plane and of x–y–z-space.
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Publisher Summary This chapter explores a particular invariant property of some mappings. Eigen is a German word meaning special, and an eigenvector of a mapping is a special vector whose direction is invariant under, or reversed by, the mapping. The chapter discusses the mappings of the x–y-plane and of x–y–z-space.
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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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