Transcription of Chapter 10 Eigenvalues and Singular Values
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Chapter 10 Eigenvalues and Singular Values
Chapter 10 Eigenvalues and SingularValuesThis Chapter is about Eigenvalues and Singular Values of matrices. Computationalalgorithms and sensitivity to perturbations are both eigenvalue and Singular Value DecompositionsAneigenvalueandeigenvector of a square matrixAare a scalar and a nonzerovectorxso thatAx= valueand pair ofsingular vectorsof a square or rectangular matrixAare a nonnegative scalar and two nonzero vectorsuandvso thatAv= u,AHu= superscript onAHstands forHermitian transposeand denotes the complexconjugate transpose of a complex matrix. If the matrix is real, thenATdenotes thesame matrix. InMatlab, these transposed matrices are denoted byA'.The term eigenvalue is a partial translation of the German eigenwert. Acomplete translation would be something like own value or characteristic value, but these are rarely used.
Consequently, the three eigenvalues are λ1 = 1, λ2 = 2, and λ3 = 3, and Λ = 1 0 0 0 2 0 0 0 3 . The matrix of eigenvectors can be normalized so that its elements are all integers: X = 1 −4 7 −3 9 −49 0 1 9 . It turns out that the inverse of X also has integer entries: X−1 = 130 43 133 27 9 …
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