### Matrix calculus: Minimal polynomial

### Minimal polynomial

Let #n# be a natural number and #A# an #(n\times n)#-matrix. If #p# is a polynomial with #p(A) = 0# (the zero matrix), then we also say that #A# is a zero of #p#. The characteristic polynomial #p_A# of an #(n\times n)#-matrix satisfies #p_A(A) = 0# and has degree #n#. But sometimes there are polynomials of lower degree of which #A# is a zero (that is: yielding the zero matrix when you substitute #A#). We recall that a polynomial is called *monic* if its leading coefficient equals #1#.

Minimal polynomial Let #n# be a natural number and #A# an #(n\times n)#-matrix.

- There is a unique monic polynomial #m_A(x)# of minimal degree such that #m_A(A) = 0#. This polynomial is called the
**minimal polynomial**of #A#. - The minimal polynomial of a matrix #A# is equal to the minimal polynomial of each conjugate of #A#. In particular, we can speak of the
**minimal polynomial**of a linear map #L:V\to V#, where #V# is an #n#-dimensional vector space, which is the minimal polynomial of the #(n\times n)#-matrix #L_\alpha# with respect to an arbitrarily chosen basis #\alpha# for #V#. We then also write #m_L# rather than #m_A#. - Each polynomial #f(x)# that satisfies #f(A) = 0# is a multiple of #m_A(x)#. In particular, #m_A# divides #p_A#.
- Each root of the characteristic polynomial of #A# is a root of the minimal polynomial of #A#.

The minimal polynomial of a square matrix #A# can be determined in at least two ways:

- Compute the characteristic polynomial #p_A#. Find for the largest monic divisor #n_A# of #p_A# without double complex roots. Search among the monic divisors of #p_A# for a multiple of #n_A# of smallest degree such that #A# is a zero of it.
- Find a linear relationship of the form #c_0\cdot I+c_1\cdot A+c_2\cdot A^2+\cdots +c_{k-1}\cdot A^{k-1}+A^k=0# for the smallest possible #k#. Then we must have #m_A (x)= c+c_1\cdot x+c_2\cdot x^2+\cdots+c_{k-1}\cdot x^{k-1}+x^k#.

The first method is feasible if #p_A# has many different roots. The second method is very straightforward. We give some examples.

We are looking for the lowest degree monic polynomial in #x# that becomes the zero matrix upon substitution of #A# for #x#. To this end, we first calculate the relevant powers of #A#:

\[\begin{array}{rcl}

A^2 &=& \matrix{-4 & -17 & 17 \\ -3 & -15 & 15 \\ -8 & -35 & 36 \\ }\\ A^3 &=& \matrix{17 & 65 & -69 \\ 15 & 60 & -63 \\ 36 & 143 & -151 \\ }

\end{array}\] We next consider the polynomial #a+b\cdot x+c\cdot x^2+d\cdot x^3# with coefficients #a#, #b#, #c#, #d# to be determined, such that the zero matrix appears after we substitute #A# for #x#. This gives

\[\matrix{17 d-4 c+a & 65 d-17 c+5 b & -69 d+17 c-4 b \\ 15 d-3 c & 60 d-15 c+3 b+a & -63 d+15 c-3 b \\ 36 d-8 c+b & 143 d-35 c+8 b & -151 d+36 c-8 b+a \\ } = \matrix{0&0&0\\ 0&0&0\\ 0&0&0} \] This is a system of #9# linear equations with unknowns #a#, #b#, #c#, #d#. Its solution, written with #d# as a parameter is

\[ a=3 d ,\phantom{xx} b=4 d ,\phantom{xx} c=5 d \] Apparently, there is a solution only if #d\ne0#. The minimal polynomial thus has degree #3#. Because the minimal polynomial #m_A(x)# is monic, we must take #d=1# to find the answer. This gives the solution \[ a=3 ,\phantom{xx} b=4 ,\phantom{xx} c=5 \] so the answer is #m_A(x) = x^3+5 x^2+4 x+3#.

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