This document discusses eigenvectors and eigenvalues. It defines an eigenvector as a non-zero vector that is mapped to a scaled version of itself by a linear transformation. This satisfies the equation Ax = λx, where λ is the eigenvalue. For a vector to be an eigenvector, the determinant of the matrix A - λI must equal 0, which is known as the characteristic equation. Solving this equation yields the eigenvalues, and corresponding eigenvectors can then be found by solving the original eigenvector equation for each eigenvalue.

1. Eigen vector
Yuji Oyamada
1 HVRL, Keio University
2 Chair for Computer Aided Medical Procedure (CAMP), Technische Universit¨t M¨nchen
a u
April 9, 2012

2. Prerequisities
Let b = Ax be a linear transformation system of n dimension.
A square matrix A is a function that transforms an n dimensional vector x
to another n dimensional vector b.
Y. Oyamada (Keio Univ. and TUM) Eigen vector April 9, 2012 2/7

3. Eigenvector
Eigen vector x is a non-zero vector that is imaged into a vector λx by A as
Ax = λx, (1)
where the scalar λ is called eigen value.
Y. Oyamada (Keio Univ. and TUM) Eigen vector April 9, 2012 3/7

4. Eigenvector (cont.)
Eq. (1) is equivalent to
λx − Ax = 0 or λIx − Ax = 0
that is
  
λ − a11 −a12 ··· −a1n x1
 −a21 λ − a22 ··· −a2n   x2 
(λI − A) x = 
 ···
   = 0. (2)
··· ··· · · ·  · · ·
−am1 −am2 ··· λ − amn xn
This equation is homogeneous!
Y. Oyamada (Keio Univ. and TUM) Eigen vector April 9, 2012 4/7

5. Homogeneous system
Homogeneous systems form as
Ax = 0.
Homogeneous systems have at least trivial solution
x=0
A non-trivial solution is any solution that
x = 0.
Homogeneous systems have non-trivial solutions if and only if the
determinant of the coefficient matrix is equal to zero as
det(A) = |A| = 0.
Y. Oyamada (Keio Univ. and TUM) Eigen vector April 9, 2012 5/7

6. Characteristic equation
Non-trivial solutoin of Eq. (2) should satisfy
det (λI − A) = |λI − A| = 0. (3)
The expansion of the equation forms as
φ(λ) = det (λI − A) = λn + cn−1 λn−1 + · · · + c1 λ + c0 . (4)
This equation is called characteristic equation of matrix A.
Y. Oyamada (Keio Univ. and TUM) Eigen vector April 9, 2012 6/7

7. Eigenvalues and eigenvectors computation
1 Compute eigenvalues {λi } by solving Eq. (3).
2 Compute eigenvector xi by solving Eq. (2) for each eigenvalue λi .
Y. Oyamada (Keio Univ. and TUM) Eigen vector April 9, 2012 7/7