OpenLLM-France/wikimedia huggingface dataset:
# Vector space/Inner product/Endomorphism/Self-adjoint/Eigentheory/Fact/Proof

* Let $v\in U^{\perp }$, and $u\in U$. Because of the invariance of $U$, we have $\varphi (u)\in U$. Therefore, This means that $\varphi (v)$ is orthogonal to $U$; thus, it belongs to $U^{\perp }$, and this means the invariance.
    $\left\langle \varphi (v),u\right\rangle =\left\langle v,\varphi (u)\right\rangle =0\,.$
* This is only relevant in case ${\mathbb {K} }=\mathbb {C}$. Let $\lambda \in \mathbb {C}$ be an eigenvalue, and let $v\in V$ be an eigenvector, that is, We may assume that this eigenvector is normed. We have therefore, $\lambda$ is real.
    $\varphi (v)=\lambda v\,.$
    $\lambda =\left\langle \lambda v,v\right\rangle =\left\langle \varphi (v),v\right\rangle =\left\langle v,\varphi (v)\right\rangle =\left\langle v,\lambda v\right\rangle ={\overline {\lambda }}\,;$
* Let $v_{1}$ be an eigenvector for the eigenvalue $\lambda _{1}$, and let $v_{2}$ be an eigenvector for the eigenvalue $\lambda _{2}\neq \lambda _{1}$. Then This is only possible for
    $\lambda _{1}\left\langle v_{1},v_{2}\right\rangle =\left\langle \lambda _{1}v_{1},v_{2}\right\rangle =\left\langle \varphi (v_{1}),v_{2}\right\rangle =\left\langle v_{1},\varphi (v_{2})\right\rangle =\left\langle v_{1},\lambda _{2}(v_{2})\right\rangle ={\overline {\lambda _{2}}}\left\langle v_{1},v_{2}\right\rangle =\lambda _{2}\left\langle v_{1},v_{2}\right\rangle \,.$
    $\left\langle v_{1},v_{2}\right\rangle =0\,.$
* We may assume that $V={\mathbb {K} }^{n}$, endowed with the standard inner product. In case ${\mathbb {K} }=\mathbb {C}$, the statement is known; so suppose that ${\mathbb {K} }=\mathbb {R}$. We may consider the mapping also as a mapping from $\mathbb {C} ^{n}$ to $\mathbb {C} ^{n}$. This map is again self-adjoint, and the characteristic polynomial does not change. Therefore, it splits into linear factors, and the zeroes are real by (2).
# Vector space/Inner product/Endomorphism/Sesquilinear form/Fact/Proof

* We have and that is, the assignment is linear in the first component, and antilinear in the second component. Therefore, $\Psi _{\varphi }$ is a sesquilinear form.
    ${\begin{aligned}\psi _{\varphi }(av_{1}+bv_{2},w)&=\left\langle \varphi (av_{1}+bv_{2}),w\right\rangle \\&=\left\langle a\varphi (v_{1})+b\varphi (v_{2}),w\right\rangle \\&=a\left\langle \varphi (v_{1}),w\right\rangle +b\left\langle \varphi (v_{2}),w\right\rangle \\&=a\psi _{\varphi }(v_{1},w)+b\psi _{\varphi }(v_{2},w)\end{aligned}}$
    ${\begin{aligned}\psi _{\varphi }(v,aw_{1}+bw_{2})&=\left\langle \varphi (v),aw_{1}+bw_{2}\right\rangle \\&={\overline {a}}\left\langle \varphi (v),w_{1}\right\rangle +{\overline {b}}\left\langle \varphi (v),w_{2}\right\rangle \\&={\overline {a}}\psi _{\varphi }(v,w_{1})+{\overline {b}}\psi _{\varphi }(v,w_{2}),\end{aligned}}$
* The linearity follows from the linearity of the inner product in the first component. In the finite-dimensional case, we have on the left-hand side and on the right-hand side vector spaces of the dimension ${\left(\dim _{K}{\left(V\right)}\right)}^{2}$; therefore, it is enough to show injectivity. If $\Psi _{\varphi }=0$ is the zero form, then $\left\langle \varphi (v),w\right\rangle =0$ for all $v,w$. In particular, $\left\langle \varphi (v),\varphi (v)\right\rangle =0$, which implies $\varphi (v)=0$.
* If $\varphi$ is not bijective, then let $v\in \operatorname {kern} \varphi$, $v\neq 0$. Then, $\Psi _{\varphi }(v,-)$ is the zero mapping in the second component, and the form is degenerate. To prove the converse, suppose that $\Psi _{\varphi }(-,-)$ is degenerate. Then there exists a vector $v\in V$, $v\neq 0$, such that $\left\langle \varphi (v),-\right\rangle$ is the zero-mapping. Since an inner product is nondegenerate, this implies $\varphi (v)=0$, and $\varphi$ is not bijective.
* In the self-adjoint case, we have The converse follows from
    $\Psi _{\varphi }(v,w)=\left\langle \varphi (v),w\right\rangle =\left\langle v,\varphi (w)\right\rangle ={\overline {\left\langle \varphi (w),v\right\rangle }}={\overline {\Psi _{\varphi }(w,v)}}\,.$
    $\left\langle \varphi (v),w\right\rangle =\Psi _{\varphi }(v,w)={\overline {\Psi _{\varphi }(w,v)}}={\overline {\left\langle \varphi (w),v\right\rangle }}=\left\langle v,\varphi (w)\right\rangle \,.$
# Linear algebra (Osnabrück 2024-2025)/Isometry/C/Diagonalizable/Fact/Factreferencenumber

Theorem 34.2
# Linear algebra (Osnabrück 2024-2025)/Vector space/Inner product/Endomorphism/Self-adjoint/Spectral theorem/Fact/Factreferencenumber

Theorem 41.11
# Normal endomorphism/Introduction/Section
## Definition

Vector space/K/Finite-dimensional/Normal endomorphism/Definition
Thus, the condition is
    $\varphi \circ {\hat {\varphi }}={\hat {\varphi }}\circ \varphi \,.$
A self-adjoint endomorphism is normal. In the case of an isometry $\varphi$, the adjoint endomorphism is $\varphi ^{-1}$, due to example. Therefore, an isometry is normal. If the endomorphism $\varphi$ is described by the matrix $M$, with respect to an orthonormal basis, then the condition for normality means
    $M{{\overline {M}}^{\text{tr}}}={{\overline {M}}^{\text{tr}}}M\,.$

## Example

The third property of the following lemma explains the word "normal“.

## Lemma
### Proof

$\Box$

## Lemma
### Proof

$\Box$

## Lemma
### Proof

$\Box$

## Lemma
### Proof

$\Box$

## Theorem
### Proof

$\Box$
# Continuous function/x sin 1 over x/On 0,1/Infinitely many zeroes/Exercise

Prove that the function
    $f(x)={\begin{cases}x\sin {\frac {1}{x}}{\text{ for }}x\in0,1]},\\0{\text{ for }}x=0,\end{cases}}\,$
is continuous, and that it has infinitely many zeros.
# Animal Phyla/Acoelomorpha

* Acoelomorpha
* Name Meaning: With gut
* English Common Name: Acoels
* Major distinguishing characteristics: No mouth or alimentary canal
* Approximate number of species described: unknown

## Natural History

All but two freshwater species are marine. All are aquatic. They live between the grains of sand, in the water column as plankton, or crawling on the substrate or other organisms.

## Taxonomy

Some consider this group to be a Subphylum of Xenacoelomorpha.
There are two Classes that traditionally belong to the Phylum regardless of their placement.
* Acoela
* Nemertodermatida

## Anatomy

The Acoelomorpha resemble flat worms. Their internal structures are so simple they have no real internal structures.

## The Fossil Record

There is only one known fossil of an acoelomorph, Gutticorpus.

## Quiz

* Acoelomorpha Quiz

## References and Further Reading

* Acoelomorpha on Encyclopedia of Life
* Acoelomorpha at Wikipedia
# Animal Phyla/Acanthocephala

* Acanthocephala
* Name Meaning: Thorny head
* English Common Name: Thorny-headed worms
* Major distinguishing characteristics: Reversible spiny proboscis
* Approximate number of species described: 1,420

## Natural History

All known Acanthocephalans are intestinal parasites of vertebrates. Some spend their early development as parasites of invertebrates, but must move to a vertebrate host in order to reproduce. They are most commonly found in fishes, but may be found in any type of vertebrate. They lodge by their "horny" (spine covered) proboscis into the intestinal wall of their host. They ingest directly through their body wall.

## Taxonomy

There are over 1,400 species described, in four Classes.
The four Classes are
* Archiacanthocephala
* Eoacanthocephala
* Palaeacanthocephala
* Polyacanthocephala

## Anatomy

Acanthocephala are a simple tubular worm with a spiny or "thorny" proboscis, which they use to embed themselves into the intestinal wall of their host.

## The Fossil Record

While no adult specimens of acanthocephalans have been found in the fossil record, their eggs have been recorded as far back as the Cretaceous.

## Quiz

* Acanthocephala Quiz

## References and Further Reading

* Acanthocephala at Encyclopedia of Life
* Acanthocephala at Tree of Life
* Classification of the Acanthocephala
* Labandeir, Conrad C. Paleobiology of Predators, Parasitoids, and Parasites: Death and Accomodation in the Fossil Record of Continental Invertebrates. in Kowalewski, M., and P.H. Kelley, (eds.), 2002. The Fossil Record of Prédation. Paleontological Society Papers, 8: 211-249. Available online at The Smithsonian Institution
* https://www.scielo.br/j/aabc/a/7YbvGFfkzr7dvFVwvYL9g8B/?lang=en
# Geochronology/Paleozoic/Quiz

Paleozoic geochronology is an article and a lecture from the school of geology.
You are free to take this quiz based on Paleozoic geochronology at any time.
To improve your score, read and study the lecture, the links contained within, listed under See also, External links, and in the {{geology resources}} template. This should give you adequate background to get 100 %.
As a "learning by doing" resource, this quiz helps you to assess your knowledge and understanding of the information, and it is a quiz you may take over and over as a learning resource to improve your knowledge, understanding, test-taking skills, and your score.
Suggestion: Have the lecture available in a separate window.
To master the information and use only your memory while taking the quiz, try rewriting the information from more familiar points of view, or be creative with association.
Enjoy learning by doing!

## Hypotheses

* Tephra layers may be the best geochronometer.
# Animal Phyla/Myxozoa

* Myxozoa
* Name Meaning: Slime animals
* English Common Name: Myxozoans
* Major distinguishing characteristics: Polar capsules derived from nematocysts
* Approximate number of species described: 13,000+
A class of Phylum Cnidaria

## Quiz

* Myxozoa Quiz

## References and Further Reading

* Myxozoa at Wikipedia
* Myxozoa at Tree of Life
* Myxozoa at Encyclopedia of Life
# Polynomials/n/Vanishing sets/Introduction/Section
## Definition

Monomial/Definition
The degree of a monomial is the sum of the exponents, that is, $\nu _{1}+\nu _{2}+\cdots +\nu _{n}$.

## Definition

Polynomial/Field/n variables/Directly/Definition
The degree of a polynomial is the maximum of the degrees of the monomials involved (meaning those monomials that occur with a coefficient different from $0$). A polynomial $F=F(X_{1},\ldots ,X_{n})$ in $n$ variables over $K$ defines a function by inserting (substituting each variable by certain values)
    $K^{n}\longrightarrow K,(x_{1},\ldots ,x_{n})\longmapsto F(x_{1},\ldots ,x_{n}).$
These are important functions in higher-dimensional analysis. The variable $X_{i}$, interpreted in this way, is simply the $i$-th projection, and the addition and the multiplication of polynomials corresponds to the addition and the multiplication of functions, where the values in $K$ are added or multiplied.

## Definition

Polynomial ring/Field/n variables/Directly/Definition

## Definition

Affine space/K/Polynomial/Vanishing set/Definition
Thus, the vanishing set of $F$ is just the fiber of the function
    $F\colon K^{n}\longrightarrow K$
given by $F$. For $n=1$, this is just a finite collection of points, the zeroes of $F$, (in case $F=0$, it is $K$); however, for $n\geq 2$, these vanishing sets are interesting and complicated geometric objects. The study of these objects is called algebraic geometry. In case of $n=2$, we talk about algebraic curves.
For arbitrary $n$, a polynomial of degree $\leq 1$ has the form
    $F=a_{1}X_{1}+\cdots +a_{n}X_{n}+b\,,$
and the corresponding vanishing set is just the solution set of the inhomogeneous linear equation
    $a_{1}x_{1}+\cdots +a_{n}x_{n}=-b\,,$
that is, it is an affine-linear space.
# Monomial/Definition

For a set of variables $X_{1},\ldots ,X_{n}$ and an $n$-tuple $(\nu _{1},\ldots ,\nu _{n})\in \mathbb {N} ^{n}$, an expression of the form $X_{1}^{\nu _{1}}\cdots X_{n}^{\nu _{n}}$ is called a monomial in the $X_{i}$.
# Polynomial/Field/n variables/Directly/Definition

A polynomial $F$ in the variables $X_{1},\ldots ,X_{n}$ over a field $K$ is a finite linear combination of monomials
    $F=\sum _{\nu }c_{\nu }X^{\nu }\,,$
where $c_{\nu }\in K$.
# Polynomial ring/Field/n variables/Directly/Definition

For a field $K$ and a set of variables $X_{1},\ldots ,X_{n}$, the polynomial ring
    $K[X_{1},\ldots ,X_{n}]$
consists of all polynomials $F(X_{1},\ldots ,X_{n})$ in these variables. This set is made into a commutative ring by componentwise addition and by the multiplication that extends the rule
    $X_{1}^{r_{1}}\cdots X_{n}^{r_{n}}\cdot X_{1}^{s_{1}}\cdots X_{n}^{s_{n}}:=X_{1}^{r_{1}+s_{1}}\cdots X_{n}^{r_{n}+s_{n}}\,$
such that distributivity holds.
# Affine space/K/Polynomial/Vanishing set/Definition

Let $K$ be a field, and let $F\in K[X_{1},\ldots ,X_{n}]$ denote a polynomial in $n$ Variables. The set
    ${\left\{P\in K^{n}\mid F(P)=0\right\}}$
is called the vanishing set (or zero set) of $F$.
# Principal axis transformation/Real quadrics/Section
## Definition

Polynomial of degree 2/K/Expanded/Definition

## Example

We develop a relation between quadratic polynomials and bilinear forms.

## Definition

Bilinear form/Corresponding quadratic form/Definition
Given a basis $v_{1},\ldots ,v_{n}$, a bilinear form is described by its Gram matrix
    $G={\left(g_{ij}\right)}_{ij}\,;$
the corresponding quadratic form $V\rightarrow K$ is described by the quadratic polynomial
    $\left(X_{1},\,\ldots ,\,X_{n}\right)G{\begin{pmatrix}X_{1}\\\vdots \\X_{n}\end{pmatrix}}=\sum _{1\leq i,j\leq n}g_{ij}X_{i}X_{j}=\sum _{i}g_{ii}X_{i}^{2}+\sum _{i<j}{\left(g_{ij}+g_{ji}\right)}X_{i}X_{j}\,.$
($X_{i}$ is the $i$-th projection, the corresponding dual basis). In the symmetric case, this is
    $\sum _{i}g_{ii}X_{i}^{2}+\sum _{i<j}2g_{ij}X_{i}X_{j}.$
For every pure-quadratic polynomial in $n$ variables, we can form in this way a symmetric Gram matrix. The theory of real-symmetric bilinear forms makes it possible to get rid, by a suitable coordinate transformation (a base change) of the mixed terms.

## Example

The polynomial $XY-1$ does not appear directly in this list, because, in the variables $X=U+V$ and $Y=U-V$, we can write
    $U^{2}-V^{2}=1\,.$
In this form it is in the lists. The following theorem tells us that, up to scaling, the list is complete.

## Theorem
### Proof

$\Box$
The representations occurring in this theorem are called the standard form of the quadratic form. In a standard form, we only have purely-quadratic terms and at most one variable in first degree. The theorem tells us that every quadratic form can be brought, using suitable orthonormal coordinates, into such a standard form. Regarding the vanishing set, such a coordinate transformation means that we apply an affine-linear isometry.
# Polynomial of degree 2/K/Expanded/Definition

A quadratic polynomial $F\in K[X_{1},\ldots ,X_{n}]$ over a field $K$ is a polynomial of degree $2$; that is, an expression of the form
    $F=\sum _{i\leq j}a_{ij}X_{i}X_{j}+\sum _{i=1}^{n}b_{i}X_{i}+c\,,$
where $a_{ij},b_{i},c\in K$.
# Polynomial of degree 2/R/1 variable/Completing the square/Example

For a quadratic polynomial $aX^{2}+bX+c$ in one variable $X$, with $a,b,c\in K$ and $a\neq 0$, the zeroes can be found by completing the square. That is, we write (suppose that the characteristic of the fields is not $2$)
    $aX^{2}+bX+c=a{\left(X^{2}+{\frac {b}{a}}X+{\frac {c}{a}}\right)}=a{\left({\left(X+{\frac {b}{2a}}\right)}^{2}-{\frac {b^{2}}{4a^{2}}}+{\frac {c}{a}}\right)}\,.$
This equals $0$ if and only if
    $X=\pm {\sqrt {{\frac {b^{2}}{4a^{2}}}-{\frac {c}{a}}}}-{\frac {b}{2a}}\,,$
and if the square root
    ${\sqrt {{\frac {b^{2}}{4a^{2}}}-{\frac {c}{a}}}}={\frac {1}{2a}}{\sqrt {b^{2}-4ac}}\,$
exists in the field. Depending on this, there are no, one or two solutions.
# Bilinear form/Corresponding quadratic form/Definition

For a bilinear form $\left\langle -,-\right\rangle$ on a $K$-vector space $V$, the mapping
    $V\longrightarrow K,v\longmapsto \left\langle v,v\right\rangle$
is called the corresponding quadratic form.
# Linear algebra (Osnabrück 2024-2025)/Vector space/Dual basis/Base change/Fact/Factreferencenumber

Lemma 14.14
# Quadratic polynomial/R/3 variables/Pure form/All variables/List/Example

We make a list of real quadratic polynomials in the three variables $X,Y$ and $Z$, together with their corresponding vanishing sets, where we restrict the coefficients to $0,1,-1$. Moreover, we consider only such polynomials where all variables occur and the vanishing set is not empty.
# Quadratic polynomial/R/2 variables/Pure form/List/Example

We make a list of real quadratic polynomials in the two variables $X$ and $Y$, together with their corresponding vanishing sets; we restrict to coefficients from $0,1,-1$. If only one variable $X$ occurs, then we have essentially the following three possibilities.
In these cases, the vanishing set is simply the product set of a zero-dimensional vanishing set (finitely many points) and of a line.
Now we consider polynomials where both variables occur.
# Quadratic polynomial/R/Change of variables/Pure form/Fact

Every real quadratic polynomial
    $F=\sum _{i\leq j}a_{ij}X_{i}X_{j}+\sum _{i=1}^{n}b_{i}X_{i}+c\,$
orthonormal basis (allowing translations), the form ($k\geq n$)
    $F=\sum _{1\leq i\leq k}r_{i}U_{i}^{2}+s\,,$
or the form ($k\leq n-1$)
    $F=\sum _{1\leq i\leq k}r_{i}U_{i}^{2}+sU_{k+1}\,.$
# Quadratic polynomial/R/Change of variables/Pure form/Fact/Proof

We consider the square matrix
    $M={\left(\alpha _{ij}\right)}_{1\leq i,j\leq n}\,,$
where
    $\alpha _{ij}={\begin{cases}a_{ij}{\text{ for }}i=j\,,\\{\frac {a_{ij}}{2}}{\text{ for }}i<j\,,\\{\frac {a_{ji}}{2}}{\text{ for }}i>j\,.\end{cases}}\,$
With this, the pure-quadratic term of the polynomial has the form
    $\left(X_{1},\,\ldots ,\,X_{n}\right)M{\begin{pmatrix}X_{1}\\\vdots \\X_{n}\end{pmatrix}}.$
This equation holds upon inserting any element from $K$ for $X_{i}$, and also as an equation in $K[X_{1},\ldots ,X_{n}]$. By definition, this matrix $M$ is symmetric. Because of fact, there exists a orthonormal basis $v_{1},\ldots ,v_{n}$ of $\mathbb {R} ^{n}$ such that the new Gram matrix
    ${B^{\text{tr}}}MB$
(describing the form with respect to the new basis, $B$ denotes the base change matrix) has diagonal form. Let $V_{1},\ldots ,V_{n}$ be the new variables with respect to the new orthonormal system; that is, the $V_{i}$ describe, considered as functions, the linear forms to this new basis, that is, the dual basis. In the new variables, there are no mixed quadratic terms any more; the polynomial has now the form
    $F=\sum _{1\leq i\leq k}e_{i}V_{i}^{2}+\sum _{j=1}^{n}f_{j}V_{j}+g\,,$
with a certain $k$ between $1$ and $n$, and $e_{i}\neq 0$. The summands
    $e_{i}V_{i}^{2}+f_{i}V_{i}$
can be brought, by completing the square and using new variables $U_{i}=V_{i}+h_{i}$, to the form
    $e_{i}U_{i}^{2}+g_{i}.$
Besides the pure-quadratic term, either a constant or a linear polynomial remains. In the second case, we denote this linear form by $U_{k+1}$.
# Quadratic polynomial/R/2 variables/Change of variables/1/Example

We consider the quadratic polynomial
    $F=3X^{2}-4XY+5Y^{2}+6X+2Y-7\,.$
We have to diagonalize the matrix
    $M={\begin{pmatrix}3&-2\\-2&5\end{pmatrix}}\,.$
The characteristic polynomial is
    $(X-3)(X-5)-4=X^{2}-8X+11=(X-4)^{2}-5\,.$
Therefore, the eigenvalues are
    $x_{1}={\sqrt {5}}+4{\text{ and }}x_{2}=-{\sqrt {5}}+4.$
Eigenvectors are
    ${\begin{pmatrix}-2\\{\sqrt {5}}+1\end{pmatrix}}\,\,{\text{ and }}\,\,{\begin{pmatrix}2\\{\sqrt {5}}-1\end{pmatrix}}.$
Hence,
    ${\frac {1}{\sqrt {10+2{\sqrt {5}}}}}{\begin{pmatrix}-2\\{\sqrt {5}}+1\end{pmatrix}}\,\,{\text{ and }}\,\,{\frac {1}{\sqrt {10-2{\sqrt {5}}}}}{\begin{pmatrix}2\\{\sqrt {5}}-1\end{pmatrix}}.$
is an orthonormal basis consisting of eigenvectors.
# Quadratic polynomial/R/Variable set/Pure form/Normed standard form/Remark

A quadratic form in standard form
    $\sum _{1\leq i\leq k}r_{i}U_{i}^{2}+s\,\,{\text{ or }}\,\,\sum _{1\leq i\leq k}r_{i}U_{i}^{2}+sU_{k+1}$
in the sense of fact can be simplified further, if we allow distortions. In the new coordinates
    $Z_{i}={\sqrt {\vert {r_{i}}\vert }}U_{i}\,$
or
    $U_{i}={\frac {1}{\sqrt {\vert {r_{i}}\vert }}}Z_{i}\,$
for $i=1,\ldots ,k$, the quadratic form has a representation of the form
    $\sum _{1\leq i\leq k}\pm Z_{i}^{2}+s\,\,{\text{ or }}\,\,\sum _{1\leq i\leq k}\pm Z_{i}^{2}+sZ_{k+1},$
the coefficients are $1$ or $-1$. This is called the normed standard form of the quadratic form. By swapping of the variables, we may achieve that the first variables have the coefficient $1$, and the later variables have coefficient $-1$. In doing these transformations, the vanishing sets are distorted. For example, an ellipse might become a circle, or a parabola might be compressed. Since the vanishing set does not change by multiplying the form with $-1$, we may also assume that the number variables with coefficient $1$ is at least the number of variables with coefficient $-1$.
# Linear algebra (Osnabrück 2024-2025)/Quadratic polynomial/R/Change of variables/Pure form/Fact/Factreferencenumber

Theorem 43.9
# Linear mappings/Commuting/Definition

We say that linear mappings
    $\varphi ,\psi \colon V\longrightarrow V$
on a $K$-vector space commute if
    $\psi \circ \varphi =\varphi \circ \psi \,$
holds.
# Vector space/K/Finite-dimensional/Normal endomorphism/Definition

Let $V$ be a finite-dimensional ${\mathbb {K} }$-vector space, endowed with an inner product $\left\langle -,-\right\rangle$. An endomorphism
    $\varphi \colon V\longrightarrow V$
is called normal if $\varphi$ and its adjoint endomorphism ${\hat {\varphi }}$ commute.
# Normal matrix/2x2/Explicit/Example

In the two-dimensional situation, the condition for normality is
    ${\begin{pmatrix}a&b\\c&d\end{pmatrix}}{\begin{pmatrix}{\overline {a}}&{\overline {c}}\\{\overline {b}}&{\overline {d}}\end{pmatrix}}={\begin{pmatrix}{\overline {a}}&{\overline {c}}\\{\overline {b}}&{\overline {d}}\end{pmatrix}}{\begin{pmatrix}a&b\\c&d\end{pmatrix}}\,;$
looking at the entries, this translates to the conditions
    $b{\overline {b}}=c{\overline {c}}\,$
and
    $a{\overline {c}}+b{\overline {d}}={\overline {a}}b+{\overline {c}}d\,.$
Besides diagonal matrices and rotation matrices, also real matrices of the form
    ${\begin{pmatrix}a&b\\-b&a\end{pmatrix}}$
have this property.
# Diagonal matrix/Orthogonal basis/Normal/Example

Suppose that the linear mapping
    $\varphi \colon {\mathbb {K} }^{n}\longrightarrow {\mathbb {K} }^{n}$
has an orthonormal basis (with respect to the standard inner product) $u_{1},\ldots ,u_{n}$ consisting of eigenvectors; that means that the describing matrix is in diagonal form
    ${\begin{pmatrix}\lambda _{1}&0&\cdots &\cdots &0\\0&\lambda _{2}&0&\cdots &0\\\vdots &\ddots &\ddots &\ddots &\vdots \\0&\cdots &0&\lambda _{n-1}&0\\0&\cdots &\cdots &0&\lambda _{n}\end{pmatrix}}.$
Then the adjoint endomorphism is described, due to example, by the complex-conjugated matrix
    ${\begin{pmatrix}{\overline {\lambda }}_{1}&0&\cdots &\cdots &0\\0&{\overline {\lambda }}_{2}&0&\cdots &0\\\vdots &\ddots &\ddots &\ddots &\vdots \\0&\cdots &0&{\overline {\lambda }}_{n-1}&0\\0&\cdots &\cdots &0&{\overline {\lambda }}_{n}\end{pmatrix}}.$
These two matrices commute, that is, we have a normal endomorphism.
# Vector space/K/Finite-dimensional/Normal endomorphism/Characterization with norm/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. Then the following properties are equivalent.
* $\varphi$ is normal.
* For all $v,w\in V$, the identity holds.
    $\left\langle {\hat {\varphi }}(v),{\hat {\varphi }}(w)\right\rangle =\left\langle \varphi (v),\varphi (w)\right\rangle \,$
* For all $v\in V$, the identity holds.
    $\Vert {{\hat {\varphi }}(v)}\Vert =\Vert {\varphi (v)}\Vert \,$