Eigenvalues and Eigenvectors - PowerPoint

							Eigenvalues and
  Eigenvectors
   Regina Lunkes
       History of Matrix Theory
   Traces seen as early as 4th Century B.C.
   Babylonians and Chinese used methods
    similar to Gauss to solve simultaneous
    linear equations
   In the 16th and 17th Centuries, many
    mathematicians developed matrix theory
    without realizing it.
    • For example, De Witt in his 1660 “Elements of
      Curves” essentially diagonalized a symmetric
      matrix, but did not think in these terms.
           History Continued
 1683: Idea of a Determinant appears
  in both Japan (Seki) and Europe
  (Leibniz)
 Other important figures:

    • Cramer
    • LaGrange
    • Gauss – “Disquisitiones Arithmeticae”
         Finally… Eigenvalues
   Cauchy- 1812
    • Uses determinants in modern sense
    • Fixes old errors in certain theorems
    • Most complete early work on
      determinants
    • Used “tableau” for “matrix of
      coefficients”
    • Found eigenvalues and gave results for
      diagonalization of a matrix
    • Introduced the idea of similar matrices,
      but not the term
    • Modern Mathematicians continued to
      tweak and develop these theories
               Why Eigen??
   The German word Eigen was given to this
    concept by Hilbert in 1904
   Eigen can be translated to “own,”
    “peculiar to,” “characteristic,” or
    “individual”
   Before being referred to as eigenvalues or
    vectors, mathematicians called them
    characteristic values and characteristic
    vectors
                   Definition
   Let A be an nxn matrix. A scalar λ is called an
    eigenvalue of A if there exists a nonzero
    vector x in R such that Ax= λx. The vector x
    is called an eigenvector corresponding to λ.
   To solve,

             Ax – λx = 0
             (A – λIn)x = 0

   Example: Find the eigenvalues and
    eigenvectors of a matrix
                 Definition
   Eigenspace: Let A be an nXn matrix and λ
    an eigenvalue of A. The set of all
    eigenvectors corresponding to λ, together
    with zero vector, is a subspace of Rn.
    This subspace is the eigenspace of λ.
   Geometric interpretation: An eigenvector
    of A is thus a vector whose direction is
    unchanged or reversed when multiplied by
    A.
          Why are they useful?
   Used in Demography, or the study of
    distribution, density, and vital statistics of
    populations
   Physicists use them to calculate axes of
    intertia
   Also appears in study of vibrations,
    electrical systems, genetics, chemical
    reactions, mechanical stress, economics,
    and biology, to name a few
         More Application
 Weather Prediction:
 Research found 117 wet days out of
  195 rainy season days, 80 dry days
  out of 615 dry season days
 Using this data, we find that the
  Long-range weather forecast for this
  weather station is:
     .25 probability wet
     .75 probability dry
         Images




Cauchy    Gauss

                  Cramer