Compute the rational canonical form of a given matrix

Compute the rational canonical form of the following matrices.

A = \begin{bmatrix} 0 & -1 & -1 \\ 0 & 0 & 0 \\ -1 & 0 & 0 \end{bmatrix}
B = \begin{bmatrix} c & 0 & -1 \\ 0 & c & 1 \\ -1 & 1 & c \end{bmatrix}
C = \begin{bmatrix} 422 & 465 & 15 & -30 \\ -420 & -463 & -15 & 30 \\ 840 & 930 & 32 & -60 \\ -140 & -155 & -5 & 12 \end{bmatrix}

Consider the matrix A. First, we use elementary row and column operations to put xI - A in Smith Normal Form. Evidently the following sequence of ERCOs works:

  1. R_1 - xR_3 \rightarrow R_1
  2. R_1 \leftrightarrow R_3
  3. C_3 - xC_1 \rightarrow C_3
  4. R_2 - xR_3 \rightarrow R_2
  5. C_3 - (1-x^2)C_2 \rightarrow C_3
  6. R_2 \leftrightarrow R_3

The resulting matrix is \begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & x^3-x \end{bmatrix}. Thus the only invariant factor of A is x^3-x, and so the rational canonical form of A is \begin{bmatrix} 0 & 0 & 0 \\ 1 & 0 & 1 \\ 0 & 1 & 0 \end{bmatrix}.

Now consider the matrix B. Again, we use ERCOs to get xI-B in Smith Normal Form. Evidently the following sequence of operations suffices.

  1. R_1 - (x-c)R_3 \rightarrow R_1
  2. C_2 + C_1 \rightarrow C_1
  3. C_3 - (x-c)C_1 \rightarrow C_3
  4. C_2 + (x-c)C_3 \rightarrow C_2
  5. R_1 + (1-(x-c)^2)R_2 \rightarrow R_1
  6. R_1 \leftrightarrow R_3
  7. C_2 \leftrightarrow C_3
  8. -C_2 \rightarrow C_2
  9. -C_3 \rightarrow C_3

Resulting in the matrix \begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & x^3 - 3cx^2 - (2-3c^2)x - c(c^2-2) \end{bmatrix}. The corresponding rational canonical form is then \begin{bmatrix} 0 & 0 & c^3-2c \\ 1 & 0 & 2-3c^2 \\ 0 & 1 & 3c \end{bmatrix}.

Finally, consider the matrix C. Again we use ERCOs to get xI-C in Smith Normal Form. Evidently the following sequence of operations suffices:

  1. R_2 - 3R_4 \rightarrow R_2
  2. R_3 + 6R_4 \rightarrow R_3
  3. R_1 + 3R_4 \rightarrow R_1
  4. \frac{1}{140}R_4 \rightarrow R_4
  5. R_1 - (x-2)R_4 \rightarrow R_1
  6. C_2 - \frac{155}{140}C_1 \rightarrow C_2
  7. C_3 - \frac{5}{140}C_1 \rightarrow C_3
  8. C_4 - \frac{x-2}{140}C_1 \rightarrow C_4
  9. \frac{155}{140}R_2 + R_1 \rightarrow R_1
  10. 3C_2 + C_4 \rightarrow C_4
  11. \frac{5}{140}R_3 + R_1 \rightarrow R_1
  12. -6C_3 + C_4 \rightarrow C_4
  13. R_1 \leftrightarrow R_4
  14. -140R_4 \rightarrow R_4

Resulting in the matrix \begin{bmatrix} 1 & 0 & 0 & 0 \\ 0 & x-2 & 0 & 0 \\ 0 & 0 & x-2 & 0 \\ 0 & 0 & 0 & (x-2)(x+3) \end{bmatrix}. The rational canonical form of C is thus \begin{bmatrix} 2 & 0 & 0 & 0 \\ 0 & 2 & 0 & 0 \\ 0 & 0 & 0 & 6 \\ 0 & 0 & 1 & -1 \end{bmatrix}.

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