Fuss–Catalan number

In mathematics, in linear algebra, the cyclic decomposition theorem is an assertion of a certain property of finite-dimenstional vector spaces in relation to linear transformations of the spaces. The theorem states that given a linear transformation of a finite-dimensional vector space over an algebraically closed field, the vector space can be expressed as a direct sum of subspaces each of which is invariant under the transformation and is cyclically generated by the transformation. This result is considered to be "one of the deepest results in linear algebra".

Preliminaries

A knowledge of certain concepts and terminology related to linear transformations is an essential prerequisite for stating and understanding the cyclic decomposition theorem and its proof. To explain these, let $T$ be a linear operator on a finite-dimensional vector space $V$ over a field $F$. Let $v$ be a vector in $V$.

Cyclic subspace

The subspace of $V$ generated by the set $\{v,Tv,T^{2}v,\dots ,T^{k}v,\dots \}$ is called the $T$-cyclic subspace generated by $v$. This subspace is denoted by $Z(v;T)$.

Annihilator of a vector

Let $F[x]$ be the ring of all polynomials in $x$ over the field $F$. The set of all polynomials $g(x)$ in $F[x]$ such that $g(T)v=0$ is called the $T$-annihilator of $v$. It is denoted by $M(v;T)$. $M(v;T)$ is an ideal in the ring $F[x]$. The ideal $M(v;T)$ consists of all multiples by elements of $F[x]$ of some fixed monic polynomial in $F[x]$. This fixed monic polynomial is denoted by $p_v(x)$ and it is also sometimes referred to as the $T$-annihilator of $v$.

Conductor

Let $W$ be subspace of $V$ which is invariant under $T$. Let $F[x]$ be the ring of all polynomials in $x$ over the field $F$. The set of all polynomials $g(x)$ in $F[x]$ such that $g(T)v\in W$ is called the $T$-conductor of $v$ into $W$ and is denoted by $S_T(v;W)$. $S_T(v;W)$ is an ideal in the ring $F[x]$. The unique monic polynomial $g(x)$ of least degree such that $g(T)v\in W$, which is the generator of the ideal $S_T(v;W)$, is also called the $T$-conductor of $v$ into $W$.

Admissible subspace

Let $W$ be a linear subspace of $V$. $W$ is called a $T$-admissible subspace of $V$ if the following conditions are satisfied.

1. $W$ is invariant under $T$; that is, for any $w$ in $W$, the vector $Tw$ is in $W$.
2. For any polynomial $f(x)$ in $F[x]$ and any vector $v$ in $V$, if $f(T)v$ is in $W$ then there is a vector $w$ in $W$ such that $f(T)v=f(T)w$.

Cyclic decomposition theorem

Let $T$ be a linear operator on a finite-dimensional vector space $V$ and let $W_0$ be a proper $T$-admissible subspace of $V$. There exist non-zero vectors $v_1,v_2,\dots ,v_r$ in $V$ with respective $T$-annihilators $p_1(x),p_2(x),\dots ,p_r(x)$ such that

1. $V=W_0\oplus Z(v_1;T)\oplus Z(v_2;T)\oplus \cdots \oplus Z(v_r;T)$.
2. $p_k(x)$ divides $p_{k-1}(x)$ for $k=2,\dots ,r$.

Furthermore, the integer $r$ and the annihilators $p_1(x),p_2(x),\dots ,p_r(x)$ are uniquely determined by (1), (2), and the fact that no $v_k$ is 0.

References

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External links

• PlanetMath: Cyclic decomposition theorem