OpenLLM-France/wikimedia huggingface dataset:
# Vector space/K/Finite-dimensional/Normal endomorphism/Characterization with norm/Fact/Proof

We have
    $\left\langle \varphi (v),\varphi (w)\right\rangle =\left\langle v,{\hat {\varphi }}(\varphi (w))\right\rangle =\left\langle v,({\hat {\varphi }}\circ \varphi )(w))\right\rangle \,$
and, using fact (3), we also have
    $\left\langle {\hat {\varphi }}(v),{\hat {\varphi }}(w)\right\rangle =\left\langle v,\varphi ({\hat {\varphi }}(w))\right\rangle =\left\langle v,(\varphi \circ {\hat {\varphi }})(w))\right\rangle \,.$
If $\varphi$ and ${\hat {\varphi }}$ commute, then we also have
    $\left\langle {\hat {\varphi }}(v),{\hat {\varphi }}(w)\right\rangle =\left\langle \varphi (v),\varphi (w)\right\rangle \,$
for arbitrary $v,w\in V$. On the other hand, if this holds, then
    $\left\langle v,({\hat {\varphi }}\circ \varphi )(w))\right\rangle =\left\langle v,(\varphi \circ {\hat {\varphi }})(w))\right\rangle \,$
for all $v\in V$, therefore, the endomorphisms commute. Hence, (1) and (2) are equivalent. From (2) to (3) is a restriction. But also (3) implies (2), because we can express the inner product with the norm alone, according to the polarization formula.
# Vector space/K/Finite-dimensional/Endomorphism/Invariant linear subspaces/Orthogonal complement/Fact

Let $V$ be a finite-dimensional ${\mathbb {K} }$-vector space, endowed with an inner product $\left\langle -,-\right\rangle$. Let
    $\varphi \colon V\longrightarrow V$
denote an endomorphism.
linear subspace $U\subseteq V$ is $\varphi$-invariant if and only if the orthogonal complement
$U^{\perp }$
${\hat {\varphi }}$
# Vector space/K/Finite-dimensional/Endomorphism/Invariant linear subspaces/Orthogonal complement/Fact/Proof

Suppose that $U$ is invariant under $\varphi$. Let $u\in U$ and $v\in U^{\perp }$. Then
    $\left\langle u,{\hat {\varphi }}(v)\right\rangle =\left\langle \varphi (u),v\right\rangle =0\,.$
The reverse statement follows, because the situation is symmetric, due to fact (3) and fact (3).
# Linear algebra (Osnabrück 2024-2025)/Vector space/K/Inner product/Finite-dimensional/Orthogonal complement/Structural properties/Fact/Factreferencenumber

Corollary 32.13
# Linear algebra (Osnabrück 2024-2025)/Vector space/Finite-dimensional/Inner product/Adjoint endomorphism/Properties/Fact/Factreferencenumber

Lemma 41.7
# Vector space/K/Finite-dimensional/Normal endomorphism/Kernel/Fact

Let $V$ be a finite-dimensional ${\mathbb {K} }$-vector space, endowed with an inner product $\left\langle -,-\right\rangle$. Let
    $\varphi \colon V\longrightarrow V$
denote a normal endomorphism.
    $\operatorname {kern} \varphi =\operatorname {kern} {\hat {\varphi }}\,.$
# Vector space/K/Finite-dimensional/Normal endomorphism/Eigenvalues/Fact

Let $V$ be a finite-dimensional ${\mathbb {K} }$-vector space, endowed with an inner product $\left\langle -,-\right\rangle$. Let
    $\varphi \colon V\longrightarrow V$
denote a normal endomorphism. Then the following statements hold.
* $\lambda \in {\mathbb {K} }$ is an eigenvalue of $\varphi$ if and only if ${\overline {\lambda }}$ is an eigenvalue of ${\hat {\varphi }}$.
* A vector $v\in V$ is an eigenvector of the eigenvalue $\lambda$ if and only if $v$ is an eigenvector of ${\hat {\varphi }}$ of the eigenvalue${\overline {\lambda }}$.
# Vector space/K/Finite-dimensional/Normal endomorphism/Eigenvalues/Fact/Proof

Let
    $\psi :=\varphi -\lambda \operatorname {Id} _{V}\,;$
its kernel is the eigenspace of $\varphi$ for the eigenvalue $\lambda$. The adjoint endomorphism of $\psi$ is, using fact,
    ${\hat {\psi }}={\hat {\varphi }}-{\left(\lambda \operatorname {Id} _{V}\right)}^{\hat}={\hat {\varphi }}-{\overline {\lambda }}{\left(\operatorname {Id} _{V}\right)}^{\hat}={\hat {\varphi }}-{\overline {\lambda }}\operatorname {Id} _{V}\,.$
Due to exercise, $\psi$ is also normal. Therefore, due to fact, we have
    $\operatorname {Eig} _{\lambda }{\left(\varphi \right)}=\operatorname {kern} {\left(\varphi -\lambda \operatorname {Id} _{V}\right)}=\operatorname {kern} {\left({\hat {\varphi }}-{\overline {\lambda }}\operatorname {Id} _{V}\right)}=\operatorname {Eig} _{\overline {\lambda }}{\left({\hat {\varphi }}\right)}\,.$
# Linear algebra (Osnabrück 2024-2025)/Vector space/K/Finite-dimensional/Normal endomorphism/Kernel/Fact/Factreferencenumber

Lemma 42.7
# Vector space/C/Finite-dimensional/Normal endomorphism/Spectral theorem/Fact

Let $V$ be a finite-dimensional complex vector space, endowed with an inner product $\left\langle -,-\right\rangle$. Let
    $\varphi \colon V\longrightarrow V$
denote an endomorphism.
$\varphi$
normal if and only if there exists an orthonormal basis consisting of eigenvectors
$\varphi$
# Linear algebra (Osnabrück 2024-2025)/Vector space/C/Finite-dimensional/Normal endomorphism/Spectral theorem/Fact/Factreferencenumber

Theorem 42.9
# Vector space/C/Finite-dimensional/Normal endomorphism/Spectral theorem/Fact/Proof

Suppose first that $u_{1},\ldots ,u_{n}$ is an orthonormal basis of $V$, where the $u_{i}$ are eigenvectors of $\varphi$. The describing matrix $M$ is a diagonal matrix, its diagonal entries are the eigenvalues. Due to fact, the adjoint endomorphism is described by the conjugated-transposed matrix. Hence, this is also a diagonal matrix; therefore, it
commutes with $M$, and $\varphi$ is normal.
We prove the converse statement by induction over the dimension of $V$. So let $\varphi$ be normal. The one-dimensional case is clear. Because of des Fundamental theorem of algebra, there exists an eigenvector $v\in V$ of $\varphi$, and we may assume that it has norm $1$. Due to fact (2), $v$ is also an eigenvector of ${\hat {\varphi }}$. This implies by fact that $W=\mathbb {C} v^{\perp }$ is invariant under $\varphi$. The restriction of $\varphi$ to $W$ is again normal, and the induction hypothesis yields the claim.
# Linear algebra (Osnabrück 2024-2025)/Vector space/K/Finite-dimensional/Normal endomorphism/Eigenvalues/Fact/Factreferencenumber

Lemma 42.8
# Linear algebra (Osnabrück 2024-2025)/Vector space/K/Finite-dimensional/Endomorphism/Invariant linear subspaces/Orthogonal complement/Fact/Factreferencenumber

Lemma 42.6
# Linear algebra (Osnabrück 2024-2025)/Vector space/Inner product/Endomorphism/Self-adjoint/Eigentheory/Fact/Factreferencenumber

Lemma 41.10
# Sesquilinear form/Hermitian/Type/Inertia law/Fact

Let $V$ be a finite-dimensional complex vector space, endowed with an Hermitian sesquilinear form $\left\langle -,-\right\rangle$ of type $(p,q)$.
Gram matrix of $\left\langle -,-\right\rangle$ with respect to every orthogonal basis is a diagonal matrix
$p$
$q$
# Hermitian form/Type via self-adjoint endomorphism/Fact

Let $V$ be a finite-dimensional ${\mathbb {K} }$-vector space, endowed with an inner product. Let $\Psi$ denote an Hermitian form on $V$, corresponding to the self-adjoint endomorphism
    $\varphi \colon V\longrightarrow V$
in the sense of fact. Let $(p,q)$ be the type of $\Psi$.
$p$
positive eigenvalues, and $q$ is the number of negative eigenvalues of $\varphi$, where we have to take this numbers with their (algebraic or geometric)
# Linear algebra (Osnabrück 2024-2025)/Vector space/Inner product/Endomorphism/Sesquilinear form/Fact/Factreferencenumber

Lemma 41.12
# Hermitian form/Type via self-adjoint endomorphism/Fact/Proof

According to fact, the characteristic polynomial of $\varphi$ splits into real linear factors. Let $\lambda _{1},\ldots ,\lambda _{r}$ be the positive zeroes and let $\lambda _{r+1},\ldots ,\lambda _{s}$ be the negative zeroes. Due to fact, we have a direct sum decomposition
    $V=\operatorname {Eig} _{\lambda _{1}}{\left(\varphi \right)}\oplus \cdots \oplus \operatorname {Eig} _{\lambda _{r}}{\left(\varphi \right)}\oplus \operatorname {Eig} _{\lambda _{r+1}}{\left(\varphi \right)}\oplus \cdots \oplus \operatorname {Eig} _{\lambda _{s}}{\left(\varphi \right)}\oplus \operatorname {Eig} _{0}{\left(\varphi \right)}\,,$
which is orthogonal with respect to the inner product ($\operatorname {Eig} _{0}{\left(\varphi \right)}$ might be zero). For vectors $v_{i}$ and $v_{j}$ from different eigenspaces, we have
    $\Psi (v_{i},v_{j})=\left\langle \varphi (v_{i}),v_{j}\right\rangle =\left\langle \lambda _{i}v_{i},v_{j}\right\rangle =\lambda _{i}\left\langle v_{i},v_{j}\right\rangle =0\,;$
therefore, the eigenspaces are also orthogonal with respect to the form $\Psi$. For
    $v=v_{1}+\cdots +v_{r}\neq 0\,$
with $v_{i}\in \operatorname {Eig} _{\lambda _{i}}{\left(\varphi \right)}$, we have
    ${\begin{aligned}\Psi (v,v)&=\Psi (v_{1},v_{1})+\cdots +\Psi (v_{r},v_{r})\\&=\Psi _{\varphi }(v_{1},v_{1})+\cdots +\Psi _{\varphi }(v_{r},v_{r})\\&=\left\langle \varphi (v_{1}),v_{1}\right\rangle +\cdots +\left\langle \varphi (v_{r}),v_{r}\right\rangle \\&=\left\langle \lambda _{1}v_{1},v_{1}\right\rangle +\cdots +\left\langle \lambda _{r}v_{r},v_{r}\right\rangle \\&=\lambda _{1}\left\langle v_{1},v_{1}\right\rangle +\cdots +\lambda _{r}\left\langle v_{r},v_{r}\right\rangle \\&>0.\end{aligned}}$
This means that on this linear subspace, the restricted form is positive definite; hence,
    $p\geq \sum _{i=1}^{r}\dim _{K}{\left(\operatorname {Eig} _{\lambda _{i}}{\left(\varphi \right)}\right)}\,.$
If $p$ were strictly larger that this dimension, there existed a $p$-dimensional linear subspace $W\subseteq V$ such that the restriction of $\Psi$ to $W$ is positive definite. Because of fact, we have
    $W\cap {\left(\operatorname {Eig} _{\lambda _{r+1}}{\left(\varphi \right)}\oplus \cdots \oplus \operatorname {Eig} _{\lambda _{s}}{\left(\varphi \right)}\oplus \operatorname {Eig} _{0}{\left(\varphi \right)}\right)}\neq 0\,.$
This yields a contradiction, since the form $\Psi$ is negative semidefinite on the right-hand space. Therefore,
    $p=\sum _{i=1}^{r}\dim _{K}{\left(\operatorname {Eig} _{\lambda _{i}}{\left(\varphi \right)}\right)}\,.$
The argument for $q$ is the same.
# Linear algebra (Osnabrück 2024-2025)/Principal axis transformation/Hermitian/Orthonormal system/Fact/Factreferencenumber

Theorem 42.12
# Principal axis transformation/Hermitian/Orthonormal system/Fact/Proof

Due to fact (2) and fact (4), we can write
    $\Psi =\Psi _{\varphi }\,$
with some self-adjoint endomorphism
    $\varphi \colon V\longrightarrow V.$
Because of fact, there exists an orthonormal basis $v_{1},\ldots ,v_{n}$ consisting of eigenvectors ot $\varphi$ with the eigenvalues $\lambda _{i}$. For this basis, we have
    $\Psi (v_{i},v_{j})=\Psi _{\varphi }(v_{i},v_{j})=\left\langle \varphi (v_{i}),v_{j}\right\rangle =\left\langle \lambda _{i}v_{i},v_{j}\right\rangle =\lambda _{i}\left\langle v_{i},v_{j}\right\rangle \,.$
Therefore, this basis is also an orthogonal basis with respect to $\Psi$.
# Linear algebra (Osnabrück 2024-2025)/Hermitian form/Type via self-adjoint endomorphism/Fact/Factreferencenumber

Theorem 42.11
# Linear algebra (Osnabrück 2024-2025)/Isometry/Adjoint endomorphism/Example/Examplereferencenumber

Example 41.2
# Linear algebra (Osnabrück 2024-2025)/Diagonal matrix/Orthogonal basis/Adjoint endomorphism/Example/Examplereferencenumber

Example 41.4
# Linear algebra (Osnabrück 2024-2025)/Part II

Linear algebra (Osnabrück 2024-2025)/Part II/Navigation
Linear algebra is a fundamental course in mathematics. It deals with "linear structures“: linear systems of equations, vectors and their representation, bases and base change, linear mappings and matrices, determinants, eigenvalues and eigenvectors, optimal descriptions of linear mappings, affine mappings, normed vector spaces, lengths and angles, analytic geometry, isometries and symmetries, bilinear forms, more general algebraic structures like equivalence relations, groups, residue class construction, and multilinear algebra.
# Linear algebra (Osnabrück 2024-2025)/Part II/List of material

* List of definitions
* Important theorems
# Linear algebra (Osnabrück 2024-2025)/Part II/Enumeration of exercise sheets

Template:Enumeration of exercise sheets 31 - 58
# Linear algebra (Osnabrück 2024-2025)/Part II/News

In preparation
# Linear algebra (Osnabrück 2024-2025)/Inner product/K/Cauchy-Schwarz/Fact/Factreferencenumber

Theorem 31.8
# Linear algebra (Osnabrück 2024-2025)/Inner product/K/Norm/Properties/Fact/Factreferencenumber

Lemma 31.9
# Linear algebra (Osnabrück 2024-2025)/Vector space/K/Inner product/Finite-dimensional/Orthonormalization/Fact/Factreferencenumber

Theorem 32.9
# Linear algebra (Osnabrück 2024-2025)/Vector space/K/Inner product/Finite-dimensional/Orthogonal complement/Structural properties/Fact/Proof/Exercise/Exercisereferencenumber

Exercise 32.24
# Linear algebra (Osnabrück 2024-2025)/Vector space/K/Inner product/Orthonormal basis/Coefficients/Fact/Factreferencenumber

Lemma 32.8
# Linear algebra (Osnabrück 2024-2025)/K^3/Cross product/Properties/Fact/Factreferencenumber

Lemma 33.3
# Linear algebra (Osnabrück 2024-2025)/Inner product/K/Polarization identity with norm/Fact/Factreferencenumber

Lemma 31.10
# Linear algebra (Osnabrück 2024-2025)/Isometry/Several characterizations with orthonormal basis/Fact/Factreferencenumber

Lemma 33.7
# Linear algebra (Osnabrück 2024-2025)/Euclidean vector space/Isometry/Orthogonal/Fact/Factreferencenumber

Lemma 33.9
# Linear algebra (Osnabrück 2024-2025)/Determinant/Transposed matrix/Universal property/Fact/Factreferencenumber

Theorem 17.5
# Linear algebra (Osnabrück 2024-2025)/Unitary vector space/Isometry/Orthogonal complement/Fact/Factreferencenumber

Lemma 34.1
# Linear algebra (Osnabrück 2024-2025)/Euclidean plan/Proper isometry/Rotation/Fact/Factreferencenumber

Theorem 34.4
# Linear algebra (Osnabrück 2024-2025)/Euclidean plane/Improper isometry/Axis reflection/Fact/Factreferencenumber

Theorem 34.5
# Linear algebra (Osnabrück 2024-2025)/Euclidean vector space/R^3/Proper isometry/Fact/Factreferencenumber

Theorem 34.7
# Linear algebra (Osnabrück 2024-2025)/Euclidean vector space/R^3/Proper isometry/Representation/Fact/Factreferencenumber

Theorem 34.8
# Linear algebra (Osnabrück 2024-2025)/Endomorphism/Real/Two-dimensional invariant/Fact/Factreferencenumber

Lemma 34.10
# Linear algebra (Osnabrück 2024-2025)/Linear isometry/Determinant is 1 or -1/Fact/Factreferencenumber

Lemma 33.13
# Linear algebra (Osnabrück 2024-2025)/Unitary vector space/Isometry/Eigenvalues/Fact/Factreferencenumber

Theorem 33.12
# Linear algebra (Osnabrück 2024-2025)/Bilinear form/Linear forms/Nondegenerate/Fact/Factreferencenumber

Lemma 38.5
# Linear algebra (Osnabrück 2024-2025)/Part II/Lecture 50

Symmetry/Space/Generalities/Fixed points/Section
    Finite symmetry groups in the plane
We first consider the symmetry groups of the plane.
Finite symmetry group/Plane/Dihedral group/Section
    Systems of semiaxes
We consider now the three-dimensional situation.
Finite symmetry group/Systems of semiaxes/Introduction/Section
    The stabilizer
Finite symmetry group/Systems of semiaxes/Stabilizer group/Section
    Footnotes
# Commensurability/Definition

Two line segments $s$ und $t$ are called commensurable, if there exists a line segment $g$ with the propery that both line segments are integer multiples of $g$
# Commensurability/Real numbers/Not 0/Definition

Real numbers $r,s\neq 0$ are called commensurable if ${\frac {r}{s}}$ is a rational number.
# Observation clock

An observational clock is a Wikiversity resource type. Collaborating users from around the globe observe natural cycles of various kinds and report the data. Observational clocks can also bring classrooms together with research.
Observational clocks have a number of uses.
* Learning/reference function: users can use simple identification "keys" to help recognize species. For example, a school class could try to match observed items with their classifications, and then add to the database of observations.
* Research function: research projects can be built around the raw collected data - e.g. the development of terminologies to better describe the cycles of natural phenomena.

## List of current observational clocks

* Bloom Clock - For horticulturalists. Track flower bloom times around the globe.
* Bug Clock - For entomologists.
* Fruit Clock - For those studying plants. Track fruit and seed production, logging them as unripe, ripe, or harvestable.
# Cross-cultural psychology

This psychology resource is a "stub". Learn how you can help Wikiversity to develop it.
Some of the major psychologists working in this area are John Berry, Michael Harris Bond, Pavel Boski, Fannie Cheung, Hawley, Geert Hofstede, Gustav Jahoda, Cigdem Kagitcibasi, Kwok Leung, Reddin, Segall, Peter B. Smith, Harry Triandis, Fons van de Vijver and Kuo-shu Yang. There are many others, and the founders have already trained an active group of graduate students to continue the tradition.
The principal journal has been the Journal of Cross-Cultural Psychology, created and until recently run by the International Association for Cross-Cultural Psychology (IACCP). However, cross-cultural psychology researchers also publish widely in other peer-reviewed psychology journals.
# Illustrated Companion to the Latin Dictionary/Scamnum

This article incorporates text from a publication now in the public domain: Rich, Anthony (1849). The illustrated companion to the Latin dictionary, and Greek lexicon. p. vi. OCLC 894670115. https://archive.org/details/illustratedcompa00rich.
SCAM'NUM. A bed-side step or stool (Ov. A. Am. ii. 211.), of an intermediate size between the scabellum and gradus (Varro, L. L. v. 168.), which was used when the bedstead was of a middle size, between the highest and lowest. (Isidor. Orig. xx. 11. 8.) Hence the expression (scandere lectum, means strictly to get into bed by the assistance of this contrivance. The example (Scamnum/1.1) is taken from a bas-relief; the legs upon which the stool is raised indicate the increased height, serving the purpose of an extra step, and if compared with the illustrations s. SCABELLUM, 1. and GRADUS, 1. will at once demonstrate the accurate distinctions between those three words and the objects expressed by them.
2. A foot-stool; of a higher and consequently more dignified character than the common one (scabellum, suppedaneum), consisting of a double step, so that the feet could rest at different elevations, as in the annexed illustration (Scamnum/2.1), from a marble bas-relief in which it is appropriately placed under the feet of Jupiter to indicate the majesty of the god, and the grandeur of the throne on which he sits. The epithet cavum, the hollow foot-stool, applied by Ovid (A. Am. i. 162.) to this object may be intended to describe the incavation formed by cutting away the step in front, as in the example; or to its being actually hollow underneath, like the preceding specimen.
3. A seat formed with a step below for the feet to rest upon, as in the annexed example (Scamnum/3.1) from a Pompeian painting. It is this property which, accurately speaking, constitutes the difference between a scamnum and a subsellium; though the distinction is not strictly preserved. Ov. Fast. vi. 305. Mart. v. 41.
4. In the technical language of the agricultural people, a balk, or long line of earth between two furrows left unbroken by the plough (Columell. ii. 2. 25. Plin. H. N. xviii. 49. § 2.); also a tract of the same character left between the ridges that are made with the hoe. Columell. iii. 13. 2.
5. In the technical language of land-surveyors (agrimensores), the breadth of a field, as opposed to striga, its length. Auct. R. Agrar. pp. 46. 125. 198. ed. Goes.
* Scamnum/1.1
* Scamnum/2.1
* Scamnum/3.1
# Hypothesis

A scientific hypothesis (pl. hypotheses) is an evidence-based, testable, and falsifiable (can be shown to be wrong) statement about how some aspect of the world works.
A hypothesis is usually written in the form of an if/then statement. It could also include the word "may".
A simple example of a hypothesis could be "fire requires oxygen." It is evidence-based because fire is seen to exist in air, which contains oxygen, and can be put out by water which can cut off the air supply to the fire. It is testable because you can devise a test (removing oxygen from the area of a fire) that will either affirm the hypothesis or not. It is falsifiable because if you take away oxygen from a fire, and the fire continues to burn, the hypothesis is shown to be wrong.

## Tips

* Make hypotheses as explicit, clear, and understandable to an outside reader as possible.
* Break complicated hypotheses down into sub-hypotheses.
* Each hypothesis should be able to be tested using a single test (or series of related tests).
* Hypotheses can be expressed as null and/or alternative hypotheses.
* Hypotheses can be written in sentences or paragraphs. They can also be numbered.
# Variable

A variable in statistics refers to a measurement whose value can vary. For example, the variable "speed" (e.g., of a car) can vary between 0 and the maximum speed of the car.
A variable differs from a constant, which is a fixed value.
The simplest variable is binary (e.g., 0 or 1).

## More

For people used to basic math, it may be confusing to think of a math problem that starts with 1 + a, but in algebra such combinations are common.
a stands for a number we don't know. It could be 1, 5, 7, ¾: any number you can imagine. What we call a is a variable, because unlike normal, or constant, numbers, it may vary.
A variable can be any letter. t is seldom used as a variable because it can get mixed up with +, but x is a commonly used variable. It doesn't get confused with X, the sign for multiplication, because in algebra X is not used for that purpose: * or parentheses are used instead. (2*4 and 2(4) mean the same thing as 2 X 4.)

## How is it used?

Variables are used as placeholders in problems; they stand for unknown numbers. It may be easy to figure out what the variable stands for; it may be difficult; it may be impossible.
In the problem a + b = c, c could be any number which can be made by combining two other numbers. c could be 3, as 1 + 2 = 3. c could be 72, as 3 + 69 = 72. Where a problem gives more information, the number of solutions becomes more limited. In the problems 1 + 2 = c and c - 2 = 1, there are limited values that could be true of c.
Figuring out the variable can be as simple as doing the basic math. With the problem 1 + 2 = c, we simply add 1 and 2 to figure out that the variable, c, equals 3. It can become considerably more complex. Sometimes, a solution is not possible, and the goal is to simplify or reduce the problem as far as it can be simplified. We cannot figure out what c is in the problem a + 4 - 2 + 6 = c, but we can simplify it into a + 8 = c.

### Exercises

For more information, see Solving equations.
Here are a few simple problems to help you get used to the idea.
Find the value of the variable.
Back to Pre-Algebra
# Linear mapping/Trigonalizable/Characterizations/1/Fact

Let $K$ denote a field, and let $V$ denote a finite-dimensional vector space. Let
    $\varphi \colon V\longrightarrow V$
denote a linear mapping. Then the following statements are equivalent.
* $\varphi$ is trigonalizable.
* There exists a $\varphi$-invariant flag.
* The characteristic polynomial $\chi _{\varphi }$ splits into linear factors.
* The minimal polynomial $\mu _{\varphi }$ splits into linear factors.