Eigenvalues and Eigenvectors

Part 1.  Commands related to eigenvalues and eigenvectors

Enter the matrix A .  Calculate the characteristic polynomial of A by hand before you go to the next step.

A  =  { {2, 0, 0},  {-8, 4, -6},  {8, 1, 9} };   
    A // MatrixForm

Construct the characteristic matrix λ I - A.

λ * DiagonalMatrix[{1, 1, 1}]  -  A  //  MatrixForm

Take the determinant of the characteristic matrix.  Compare with your hand calculation of the characteristic polynomial.

Det[%]

Factor the equation by hand, and determine the eigenvalues of A before you go on.  Check by asking Mathematica for the roots.

Roots[% == 0,  λ]

Now let Mathematica compute the eigenvalues directly.

Eigenvalues[A]

One of the eigenvalues of A is 6.  Let's find corresponding eigenvectors (a basis for the eigenspace) by finding the nullspace of 6*I-A.

B  =  6 * DiagonalMatrix[{1, 1, 1}] - A;  B  //  MatrixFormNullSpace[B]

Before you go on, fill in here what the last answer means.  Check your answer with an appropriate calculation.

Now enter Mathematica's  Eigenvectors command, and explain the output.

Eigenvectors[A]

P  =  Transpose[Eigenvectors[A]];  P  //  MatrixForm

Enter the following command, and explain what you think it shows.

Inverse[P] . A . P  //  MatrixForm

Part 2.  The matrix P

Enter the following line, and explain the output.

v2  =  Map[#[[2]]&,  P]   A . v2 6 * v2

Confirm your observation in the preceding step by carrying out the same operations with the first and third columns of P.

Part 3.  No real eigenvalues

Enter the matrix B.

B  =  { {1, -1},  {1, 1} };   B // MatrixForm

Part 4.  Multiplicities

Enter the matrix X and compute its eigenvalues.

\(X\  = 
      \ {\ {331, 290, \(-58\), \ 580}, \ {603, 448, \(-213\), 1005}, 

        		\ \ \ \ \ \ \ \ {\(-105\), \(-75\), 38, \(-175\)}, 
        \ {\(-498\), \(-395\), 143, \(-847\)}\ }; 
X\  // \ MatrixForm\)

Part 5.  Summary