# Eigenvalues and Eigenvectors - PowerPoint

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9/11/2012
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```							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

```
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