### 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{21 & 5 & 4 \\ -16 & -4 & -4 \\ 32 & 9 & 9 \\ }\\ A^3 &=& \matrix{100 & 25 & 21 \\ -84 & -20 & -16 \\ 173 & 41 & 32 \\ }

\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

\[\matrix{100 d+21 c+4 b+a & 25 d+5 c+b & 21 d+4 c+b \\ -84 d-16 c-4 b & -20 d-4 c+a & -16 d-4 c \\ 173 d+32 c+9 b & 41 d+9 c+b & 32 d+9 c+a \\ } = \matrix{0&0&0\\ 0&0&0\\ 0&0&0} \] This is a system of #9# linear equations with unknowns #a#, #b#, #c#, #d#. Its solution, written with #d# as a parameter is

\[ a=4 d ,\phantom{xx} b=-5 d ,\phantom{xx} c=-4 d \] 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 \[ a=4 ,\phantom{xx} b=-5 ,\phantom{xx} c=-4 \] so the answer is #m_A(x) = x^3-4 x^2-5 x+4#.

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