Eigenvalues and Eigenvectors

Part 1. Calculation of Eigenvalues and Eigenvectors

Define the three matrices A, B, and J, and the 3x3 identity matrix I3.  (C, D, and E are all reserved variables in Mathematica.)

A  =  {{2, 1, 1}, {2, 3, 4}, {-1, -1, -2}};
A // MatrixForm

B  =  {{2, -1, 1}, {0, 3, -1}, {2, 1, 3}};
B // MatrixForm

J  =  {{2, 1, 1}, {2, 3, 2}, {3, 3, 4}};
J // MatrixForm

I3  =  IdentityMatrix[3];
I3 // MatrixForm

Calculate the characteristic polynomial of A.

P  =  Det[ A - λ*I3]

Now factor the polynomial.

Factor[P]

Find an eigenvector for each eigenvalue of A.

λ  =  1;   
RowReduce[A - λ*I3]  // MatrixForm

The following Mathematica command determines eigenvalues and eigenvectors directly.

Eigensystem[A]

Using the following text cell, explain the output in terms of your previous calculations.

Part 2. Solutions of Systems of Differential Equations

Describe the solutions of each of the following systems:

(dX/dt) = AX

Answer:

(dX/dt) = BX

Answer:

(dX/dt) = CX

Answer:

Part 3. Eigenvalues and Determinants

Find a rule relating the eigenvalues of a matrix and the determinant of the matrix.

Eigenvalues[A]
Det[A]

Eigenvalues[B]
Det[B]

Eigenvalues[J]
Det[J]

Check your rule.  The following command lines create a symmetric 3x3 matrix of random integers and calculates its eigenvalues and determinant.  Note that because Mathematica does not have a built-in function to create symmetric matrices, we needed to write one ourselves.  First, we write a function to create random square matrices.  Next, we use the property that the sum of a square matrix and its transpose is a symmetric matrix.

RandomMatrix[n_] :=  
        Table[Table[Random[Integer, {-100, 100}], {j, n}], {i, 
            n}];
\(SymmetricRandomMatrix[n_] := \ 
      Module[
{A}, 
A\  = \ RandomMatrix[n]; 

        A\  = \ A\  + \ Transpose[A];
]\)

e  =  SymmetricRandomMatrix[3];
% // MatrixForm

Transpose[e] == e

N[Eigenvalues[e]]
Det[e]

Remember that the eigenvalues of real symmetric matrices are necessarily real.  If you get imaginary eigenvalues above, note that the imaginary parts are all very small.  This is just round-off error from using the N operator.  Using algebraic manipulation, the eigenvalues may be shown to be real.

Give an algebraic justification of your rule in the text cell below.

Answer:

Part 4. The Trace Operation

Examine the calculations below, and formulate a rule for the calculation of the trace of a matrix.

A // MatrixForm
Tr[A]

B // MatrixForm
Tr[B]

J // MatrixForm
Tr[J]

F  =  DiagonalMatrix[{2, 4, 3}];
F // MatrixForm
Tr[F]

Check your rule for calculating the trace.

G  =  RandomMatrix[3];
G // MatrixForm
Tr[G]

Examine the calculations below,and formulate a rule relating the trace of a matrix to the eigenvalues of the matrix.

Eigenvalues[A]
Tr[A]

Eigenvalues[B]
Tr[B]

Eigenvalues[J]
Tr[J]

Eigenvalues[F]
Tr[F]

Using both symmetric and non-symmetric matrices, check your rule connecting the trace of a matrix to the eigenvalues of the matrix.

\( (*\ Symmetric\ Case\ *) 
\(H\  = \ 
        SymmetricRandomMatrix[3];\)\)
H // MatrixForm
N[Eigenvalues[H]]
Tr[H]

\( (*\ Non - Symmetric\ Case\ *) 
\(L\  = \ RandomMatrix[3];\)\)
L // MatrixForm
N[Eigenvalues[L]]
Tr[L]

Explain in the text cell below, why your rule must be true for the special case of diagonal matrices.

Answer:

Part 5. Summary