Diagonalization

The diagonalization instructions are:

• MAD: given a square matrix, returns determinant, formal inverse, a list polynomial and the characteristic polynomial. The list polynomial P_A is a matrix coefficient polynomail defined by the relation:
  
   

  (x I_n -A)P_A(x)=M(x) I_n = M(x) I_n - M(A) (2)

  
  
  where M denotes the characteristic polynomial of A.
• PCAR: characteristic polynomial using det
• JORDAN: compute eigenvalues and eigenvectors (cf. infra)

Given a square matrix A, JORDAN returns 6 levels:

• 6: $\det(A)^{-1}$
• 5: A^-1
• 4: list of eigenvalues (with multiplicities)
• 3: characteristic polynomial
• 2: minimal polynomial M (it divides the characteristic polynomial)
• 1: list of characteristic spaces tagged by the corresponding eigenvalue (either a vector or a list of Jordan chains, each of them ending by a "Eigen:"-tagged eigenvector)

Examples:

1.

$$A=\left(\begin{array}{lcr} 1&-1 & 0\\ 0 & 1 & -1 \\ -1 & 0 & 1 \end{array}\right)$$

returns:

6: 0
5: { { inf inf inf }
     { inf inf inf }
     { inf inf inf } }
4: {0 1 '3/2+i/2*V3' 1 '3/2-i/2*V3' 1 }
3: 'X^3-3*X^2+3*X'
2: 'X^3-3*X^2+3*X'
1: {    :0: {1 1 1}
        :'3/2+i/2*V3': {1 '-1/2-i/2*V3' '-1/2+i/2*V3'}
        :'3/2-i/2*V3': {1 '-1/2+i/2*V3' '-1/2-i/2*V3'} }

This means that A has 3 eigenvalues $\frac{3\pm\sqrt{3}i }{2}$, and a basis of eigenvectors is:

$$\{ (1,1,1), (1,\frac{-1\mp i\sqrt{3} }{2}, \frac{-1\pm i\sqrt{3} }{2}) \}$$

corresponding to $0, (3+\sqrt{3}i)/2, (3-\sqrt{3}i)/2$. The characteristic and minimal polynomial are identical (this is generically the case) X³-3X²+3X. The matrix is not invertible and has a 0 determinant.

2.

For the identity matrix I₂ (2 idn), we get:

6: 1
5: { { 1 0 } { 0 1 } }
4: {1 2}
3: 'X^2-2*X+1'
2: 'X-1'
1: { :1, Eigen: { 0 1 } :1, Eigen: { 1 0 } }

The minimal polynomial is X-1, different from the characteristic polynomial (X-1)²=X²-2X+1.

3.

$$\left(\begin{array}{ccc} 1 &2 &1 \\ 2 & 0 &0 \\ 1 & 0 &3 \end{array}\right)$$

4.

An example with 1 parameter:

{ { 1 A }
  { A 1 } }

5.

In dimension greater than 2, the factorization routines may fail. For this reason, you may have to call MAD, factor the characteristic polynomial (e.g. by trying the FCTR instruction of ALG48) before calling JORDAN. If you have ALG48 installed, try this:

{ { 1 1 A }
  { 1 A 1 }
  { A 1 1 } }
MAD FCTR JORDAN

Note that this example is solved by typing JORDAN directly but it may fail in other situations.

6.

Jordan cycles example:

$$A=\left(\begin{array}{ccc} 3& -1& 1\\ 2 & 0 & 1\\ 1 & -1 & 2 \end{array}\right),$$

returns:

6: -4
5: : { { '1/4' '1/4' '-1/4' }
       { '-3/4' '5/4' '-1/4' }
       { '-1/2' '1/2' '1/2'  } }
4: { 2 2 1 1}
3: 'X^3-5*X^2+8*X-4'
2: 'X^3-5*X^2+8*X-4'
1: { :2, Char: { 2 2 1 }  :2, Eigen:{ 1 1 0 }  :1: { 0 1 1 } }

This means that 2 has multiplicity 2, but the corresponding eigenspace is only 1-dimensional (spanned by (1,1,0) the last vector of the Jordan chain). The first vector (2,2,1) is only a characteristic vector, his image by (A-2I) is the eigenvector (1,1,0) .

Remark 8   You can not use the current variable name as a parameter of a symbolic matrix that you want to diagonalize. This would lead to incorrect results. For example, is VX is set to X, you can not diagonalize the following matrix:

{ { 1 1 X }
  { 1 X 1 }
  { X 1 1 } }

Workaround: make a change of variable, e.g. 'X=A' EXEC.