Eigenvalues and Eigenvectors

Eigenvalues and eigenvectors are defined as follows: If A is an nxn matrix and if there exist vectors x and scalars c such that Ax=cx (or (A-cI)x=0 ) then c is called an eigenvalue of A and x is called an eigenvector of A associated with c. This notion is crucial in solving applied problems.

Mathematica will quickly find eigenvalues and eigenvectors of a matrix.

In[98]:=

  m={{1, 1, 1}, {0, 1, 0}, {1, 1, 1}}

Out[98]=

  {{1, 1, 1}, {0, 1, 0}, {1, 1, 1}}

In[99]:=

  Eigenvalues[m]

Out[99]=

  {0, 1, 2}

In[100]:=

  Eigenvectors[m]

Out[100]=

  {{-1, 0, 1}, {1, -1, 1}, {1, 0, 1}}

The eigenvectors are listed in order to match their eigenvalues. If an eigenvalue is listed more than once, that means it is a multiple root of the characteristic polynomial. If the zero vector is listed with the eigenvectors, that means the matrix does not have a full complement of eigenvectors. This means that if an eigenvalue is listed twice, there may not be two corresponding eigenvalues.

In[101]:=

  s={{0, 1, 0}, {0, 0, 1}, {0, 0, 0}}

Out[101]=

  {{0, 1, 0}, {0, 0, 1}, {0, 0, 0}}

In[102]:=

  Eigenvalues[s]

Out[102]=

  {0, 0, 0}

In[103]:=

  Eigenvectors[s]

Out[103]=

  {{1, 0, 0}, {0, 0, 0}, {0, 0, 0}}

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