Chapter 10 Eigenvalues and Singular Values

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Chapter 10Eigenvalues 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 …

  Eigenvalue, Eigenvectors