### Invariant subspaces of linear maps: Diagonalizability

### The greatest common divisor of two polynomials

When dealing with the notion of *minimal polynomial* we used *division with remainder for polynomials*. We have seen that polynomials have arithmetic similarities with integers, where the degree of a polynomial is comparable to the absolute value of an integer. This also holds for the notion of greatest common divisor, which we will use later on to find the *Jordan normal form*, a unique form of a square matrix within its *conjugation class*.

Greatest common divisorLet \(f,g\) be polynomials in one variable. A **common divisor** of \(f\) and \(g\) is a polynomial dividing both \(f\) and \(g\).

If \(f\) and \(g\) are both unequal to the zero polynomial, then the common divisor of the highest possible degree is called a **greatest common divisor **(gcd).

Every greatest common divisor of \(f\) and \(g\) is a multiple of every divisor of \(f\) and \(g\).

In particular, the greatest common divisors are unique up to a scalar multiple not equal to #0#. Each set of two polynomials of which at least one is not equal to #0#, hence, has a unique *monic* gcd.

With #\gcd(f,g)# we indicate the monic gcd of #f# and #g#.

The following rules are useful for the calculation of gcd's of polynomials. They show great similarities with the rules of calculation for the gcd of integers.

Rules of calculation for the gcd of polynomials

Let #f# and #g# be polynomials, of which at least one is not equal to #0#.

- #{\gcd}(f,g)={\gcd}(g,f)#.
- #{\gcd}(0,g)=\frac{g}{c}# where #c# is the leading coefficient of #g#.
- #\gcd(f,g) = \gcd(g,r)#, in which #r# is the remainder when dividing #f# by #g#.
- If #m# is a commmon divisor of #f# and #g# with leading coefficient equal to #1#, then we have #\gcd( f, g) = m\cdot\gcd\left(\frac{f}{m},\frac{g}{m}\right)#.

The gcd can be determined by use of the

*gcd rule for polynomials*regarding the replacement of an argument of the gcd by a remainder as follows:

\[\begin{array}{rcl} \gcd (f(x),g(x))&=&{\rm gcd}(x^{10}-1,x^{15}-1)\\

&=& \gcd(x^{10}-1,x^5-1)\\

&&\phantom{xx}\color{blue}{\text{second argument replaced by remainder}}\\

&&\phantom{xx}\color{blue}{\text{by dividing this argument by the first argument}}\\

&=& \gcd(0,x^5-1)\\

&&\phantom{xx}\color{blue}{\text{first argument replaced by remainder}}\\

&&\phantom{xx}\color{blue}{\text{by dividing this argument by the second argument}}\\

&=& x^5-1

\end{array}\]

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