Linear mapping/Trigonalizable/Characterizations/1/Fact/Proof

Proof

Form (1) to (2). Let ${}v_{1},\ldots ,v_{n}$ be a basis with the property that the describing matrix of ${}\varphi$ with respect to this matrix is an upper triangular form. The action of this matrix shows directly that the linear subspaces

${}V_{i}:=\langle v_{1},\ldots ,v_{i}\rangle \,$

are ${}\varphi$-invariant and that they form an invariant flag.

From (2) to (1). Let

${}0=V_{0}\subset V_{1}\subset \ldots \subset V_{n-1}\subset V_{n}=V\,$

be a ${}\varphi$-invariant flag. Due to fact, there exists a basis ${}v_{1},\ldots ,v_{n}$ of ${}V$ such that

${}V_{i}=\langle v_{1},\ldots ,v_{i}\rangle \,.$

Since this flag is invariant, we have

${}\varphi (v_{i})=b_{1i}v_{1}+b_{2i}v_{2}+\cdots +b_{ii}v_{i}\,.$

Therefore, the describing matrix of ${}\varphi$ with respect to this basis is upper triangular.

From (1) to (3). The characteristic polynomial of ${}\varphi$ is equal to the characteristic polynomial ${}\chi _{M}$, where ${}M$ denotes a describing matrix with respect to an arbitrary basis. We may assume that ${}M$ is an upper triangular matrix. Then, according to fact, the characteristic polynomial is the product of the linear factors arising from the diagonal entries.

From (3) to (4). Due to fact, the minimal polynomial divides the characteristic polynomial.

From (4) to (2). We prove the statement by induction over ${}n$, the cases

${}n=0,1\,$

being clear. Due to the condition and fact and fact, the mapping ${}\varphi$ has an eigenvalue. Because of fact, there exists an ${}(n-1)$-dimensional linear subspace

${}V_{n-1}\subset V\,$

which is ${}\varphi$-invariant. Due to fact, the minimal polynomial of the restriction ${}\varphi {|}_{V_{n-1}}$ divides the minimal polynomial of ${}\varphi$, therefore, it also splits into linear factors. By the induction hypothesis, there exists an ${}\varphi {|}_{V_{n-1}}$-invariant flag

${}V_{1}\subset V_{2}\subset \ldots \subset V_{n-2}\subset V_{n-1}\,,$

and so this is also a ${}\varphi$-invariant flag.

The supplement follows as follows. Suppose that the trigonalizable mapping ${}\varphi$ is described by the matrix ${}M$ with respect to the basis ${}{\mathfrak {u}}$, and by the upper triangular matrix ${}T$ with respect to the basis ${}{\mathfrak {v}}$ Then, because of fact the relation ${}T=BMB^{-1}$ holds, where ${}B$ is the transformation matrix of the base change.

To fact