EIGENVALUES

EIGENVALUES

1. Possess

   • Important features of their matrix
     • sum of eigenvalues equals the matrix trace
     • if there is a zero eigenvalue, then the matrix must be SINGULAR
     • Product of eigenvalues equals the matrix determinant

2. can be used to compute determinants

3. are related to

   • EIGENVECTORS
     • Comprise
       • Eigenspace
         • which is the vector space composed of every linear combination of the eigenvector corresponding to a specific eigenvalue
     • are related matrices possess interesting properties such as
       • if A is invertible, then its inverse matrix has the same eigenvectors with different associated eigenvalues 1/lamda
     • are special vectors associated with a matrix and its eigenvalues which satisfy

4. have

   • algebraic multiplicity
     • the number of times a particular eigenvalue appears in the solution set of the characteristic polynomial
   • geometric multiplicity
     • the dimension of the eigenvalue's corresponding eigenspace

5. can be

   • real or complex number
     • which solve

6. are special numbers which satisfy

   • the characteristic polynomial
     • which is

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