OpenLLM-France/wikimedia huggingface dataset:
# LMIs in Control/Click here to continue/Integral Quadratic Constraints/Quadratic Stability and IQCs
## The System

Consider the system of differential equations
    ${\dot {x}}(t)=(A+B\Delta (t)C)x(t)\quad \Delta (t)\in {\mathfrak {D}}$
where $A,B,C$ are given and $A$ is Hurwitz. ${\mathfrak {D}}$ is the set of all diagonal matrices with the norm not exceeding 1.

## The Problem

The system is called quadratically stable if there exists a matrix $P=P^{T}$ such that
$P(A+B\Delta _{i}C)+(A+B\Delta _{i}C)^{T}P<0,\quad \forall {i}$
The stability of the system above is equivalent to the stability of the feedback interconnection:
${\begin{cases}v=Gu+f,\\u=\Delta (v)+r\end{cases}}$
where $G$ is the linear time-invariant operator with transfer function $G(s)=C(sI-A)^{-1}B$, and $\Delta$ is the operator, $\Delta (t)\in {\mathfrak {D}}.$

## The Data

Let $\Pi ={\begin{bmatrix}Q&S\\S^{T}&R\end{bmatrix}}$
where $Q=Q^{T},R=R^{T},S$ are real matrices such that
$Q+S\Delta +\Delta ^{T}S^{T}+\Delta ^{T}R\Delta >0,\quad \forall {\Delta }\in {\mathfrak {D}}$
For a fixed matrix $\Pi$ satisfying the inequality above, a sufficient condition of stability is given by
${\begin{bmatrix}G(j\omega )\\I\end{bmatrix}}^{*}\Pi {\begin{bmatrix}G(j\omega )\\I\end{bmatrix}}<0\qquad \forall \omega \in \mathbb {R} \cup \{\infty \}$

## The LMI

If there exists a $P=P^{T}$ such that
${\begin{bmatrix}PA+A^{T}P+C^{T}QC&PB+C^{T}S\\B^{T}P+S^{T}C&R\end{bmatrix}}<0$
then the system given by ${\dot {x}}(t)=(A+B\Delta (t)C)x(t)\quad \Delta (t)\in {\mathfrak {D}}$ is quadratically stable.

## References

A. Megretski and A. Rantzer, "System analysis via integral quadratic constraints," in IEEE Transactions on Automatic Control, vol. 42, no. 6, pp. 819-830, June 1997, doi: 10.1109/9.587335
P. Seiler, "Stability Analysis With Dissipation Inequalities and Integral Quadratic Constraints," in IEEE Transactions on Automatic Control, vol. 60, no. 6, pp. 1704-1709, June 2015, doi: 10.1109/TAC.2014.2361004
# LMIs in Control/Click here to continue/LMIs in system and stability Theory/Algebraic Riccati Inequalities

Algebraic Riccati Equations are particularly significant in Optimal Control, filtering and estimation problems. The need to solve such equations is common in the analysis and linear quadratic Gaussian control along with general Control problems. In one form or the other, Riccati Equations play significant roles in optimal control of multivariable and large-scale systems, scattering theory, estimation, and detection processes. In addition, closed forms solution of Riccti Equations are intractable for two reasons namely; one, they are nonlinear and two, are in matrix forms.

## The System

    ${\begin{aligned}{\dot {x}}(t)&=Ax(t)+Bu(t)\\\end{aligned}}$

## The Data

Following matrices are needed as Inputs:.
    $A,B,N$.

## The Optimization Problem

In control systems theory, many analysis and design problems are closely related to Riccati algebraic equations or inequalities. Find

## The LMI: Algebraic Riccati Inequality

Title and mathematical description of the LMI formulation.
       P>0, Q>0, R>0. The Algebraic Riccati inequality is given byA$^T$P+PA−(PB+N$^T$)R$^−$1(B$^T$P+N)+Q≥0 can be written using the Schur complement lemma as[A$^T$P+PA+QPB+N$^T$⋆R]≥0{\textstyle {\begin{aligned}P>0, \\Q>0, \\R>0.\\{\text{ The Algebraic Riccati inequality is given by}}\;\\A^{T}P+PA-(PB+N^{T})R^{-}1(B^{T}P+N)+Q&\geq 0\\\\{\text{ can be written using the Schur complement lemma as}}\;\\{\begin{bmatrix}A^{T}P+PA+Q&&PB+N^{T}\\\star &&R\\\end{bmatrix}}&\geq 0\end{aligned}}}

## Conclusion:

If the solution exists, LMIs give a unique, stabilizing, symmetric matrix P.

## Implementation:

Matlab code for this LMI in the Github repository:
* REDIRECT- CODE

## External links

* -Optimal Solution to Matrix Riccati Equation
* https://https://arxiv.org/abs/1903.08599/ LMI Properties and Applications in Systems, Stability, and Control Theory.- - A List of LMIs by Ryan Caverly and James Forbes
# LMIs in Control/Click here to continue/LMIs in system and stability Theory/Circular Region D-Admissibility
## The System

Consider $A,E\in \mathbb {R} ^{n\times n},a,b\in \mathbb {R}$ and $d\in \mathbb {R} _{>0},$ where $b\neq 0.$ The pair $(E,A)$ is D-admissible with $D={z\in \mathbb {C} :a+2bRe(z)+d\vert z\vert ^{2}<0}$ if and only if there exist $X\in \mathbb {R} ^{n\times n}$ and $\alpha \in \mathbb {R}$ such that $E^{T}X=X^{T}E\geq 0$ and

## The LMI (Constraint)

where $E^{\dagger }$ is the pseudoinverse of $E$. The region D describes a circular region of the complex plane with radius $r={\sqrt {-a/d+b^{2}/d^{2}}}$ centered at $(c,0),$ where $c=-b/d.$

## Reference

Caverly, Ryan; Forbes, James (2021). LMI Properties and Applications in Systems, Stability, and Control Theory.
# LMIs in Control/Click here to continue/LMIs in system and stability Theory/Circular Region Stability via the Dilation Lemma
## Definition

Consider $A\in \mathbb {R} ^{n\times n}$. The matrix $A$ is ${\mathcal {D}}$-stable if and only if there exists $P\in \mathbb {S} ^{n}$, where $P>0$, such that
$[\lambda _{kl}P+\phi _{kl}AP+\phi _{lk}PA^{T}]_{1<k,l<m}<0$,
or equivalent
$\Lambda \otimes P+\Phi \otimes (AP)+\Phi ^{T}\otimes (PA^{T})<0$,
where $\otimes$ is the Kroenecker product,
The eigenvalues of a ${\mathcal {D}}$-stable matrix lie within the LMI region ${\mathcal {D}}$, which is defined as
${\mathcal {D}}=\{z\in \mathbb {C} :f_{D}(z)<0\}$, where
$f_{D}(z):=\Lambda +z\Phi +{\bar {z}}\Phi ^{T}=\{\lambda _{kl}+\phi _{kl}z+\phi _{lk}{\bar {z}}\}_{1\leq k,l\leq m}$,
$\Lambda \in \mathbb {S} ^{m}$, $\Phi \in \mathbb {R} ^{m\times m}$, and ${\bar {z}}$ is the complex conjugate of $z$.

## Circular Region Stability via the Dilation Lemma

Consider $A\in \mathbb {R} ^{n\times n}$, $r\in \mathbb {R} _{>0}$, and $c\in \mathbb {R} _{<0}$, where $c<-r$. The matrix $A$ satisfies $\lambda (A)\subset {\mathcal {G}}(c,r)$, where ${\mathcal {G}}(c,r):=\{\lambda \in \mathbb {C} :\left\vert \lambda -c\right\vert <r\}$, if and only if there exist $X\in \mathbb {S} ^{n}$ and $\epsilon \in \mathbb {R} _{>0}$, where $X>0$, such that
$AX+XA^{T}-{\frac {c^{2}-r^{2}}{c}}X-{\frac {1}{c}}AXA^{T}<0$.
Equivalently, the matrix $A$ satisfies $\lambda (A)\subset {\mathcal {G}}(c,r)$ if and only if there exist $X\in \mathbb {S} ^{n}$, $\epsilon \in \mathbb {R} _{>0}$, and $F\in \mathbb {R} ^{n\times n}$, where $X>0$, such that
${\begin{bmatrix}0&-X&X&0\\*&0&0&-X\\*&*&{\frac {c}{c^{2}-r^{2}}}X&0\\*&*&*&cX\end{bmatrix}}+He{\begin{Bmatrix}{\begin{bmatrix}A\\1\\0\\0\end{bmatrix}}F{\begin{bmatrix}1&-\epsilon 1&\epsilon 1&1\end{bmatrix}}\end{Bmatrix}}<0$
Moreover, for every $X$ that satisfies
$AX+XA^{T}-{\frac {c^{2}-r^{2}}{c}}X-{\frac {1}{c}}AXA^{T}<0$
$X$ and $F=-\epsilon ^{-1}(A-\epsilon ^{-1}1)^{-1}X$ are solutions to
${\begin{bmatrix}0&-X&X&0\\*&0&0&-X\\*&*&{\frac {c}{c^{2}-r^{2}}}X&0\\*&*&*&cX\end{bmatrix}}+He{\begin{Bmatrix}{\begin{bmatrix}A\\1\\0\\0\end{bmatrix}}F{\begin{bmatrix}1&-\epsilon 1&\epsilon 1&1\end{bmatrix}}\end{Bmatrix}}<0$

## External Links

* LMI Methods in Optimal and Robust Control - A course on LMIs in Control by Matthew Peet.
* LMIs in Systems and Control Theory - A downloadable book on LMIs by Stephen Boyd.
* LMI Properties and Applications in Systems, Stability, and Control Theory - A downloadable book on LMIs by Ryan James Caverly and James Richard Forbes.
# LMIs in Control/Click here to continue/LMIs in system and stability Theory/Conic Sector Region Stability via the Dilation Lemma
## Definition

Consider $A\in \mathbb {R} ^{n\times n}$. The matrix $A$ is ${\mathcal {D}}$-stable if and only if there exists $P\in \mathbb {S} ^{n}$, where $P>0$, such that
$[\lambda _{kl}P+\phi _{kl}AP+\phi _{lk}PA^{T}]_{1<k,l<m}<0$,
or equivalent
$\Lambda \otimes P+\Phi \otimes (AP)+\Phi ^{T}\otimes (PA^{T})<0$,
where $\otimes$ is the Kroenecker product,
The eigenvalues of a ${\mathcal {D}}$-stable matrix lie within the LMI region ${\mathcal {D}}$, which is defined as
${\mathcal {D}}=\{z\in \mathbb {C} :f_{D}(z)<0\}$, where
$f_{D}(z):=\Lambda +z\Phi +{\bar {z}}\Phi ^{T}=\{\lambda _{kl}+\phi _{kl}z+\phi _{lk}{\bar {z}}\}_{1\leq k,l\leq m}$,
$\Lambda \in \mathbb {S} ^{m}$, $\Phi \in \mathbb {R} ^{m\times m}$, and ${\bar {z}}$ is the complex conjugate of $z$.

## Conic Sector Region Stability via the Dilation Lemma

Consider $A\in \mathbb {R} ^{n\times n}$ and $k\in \mathbb {R} _{>0}$.
The matrix $A$ satisfies $\lambda (A)\subset {\mathcal {P}}(k)$, where ${\mathcal {P}}(k):=\{\lambda \in \mathbb {C} :\left\vert Im(\lambda )\right\vert <k\left\vert Re(\lambda )\right\vert \}$, if and only if there exist $X\in \mathbb {S} ^{n}$ and $\epsilon \in \mathbb {R} _{>0}$, where $X>0$, such that
${\begin{bmatrix}k(AX+XA^{T})&AX-XA^{T}\\*&k(AX+XA^{T})\end{bmatrix}}<0$.
Equivalently, the matrix $A$ satisfies $\lambda (A)\subset {\mathcal {P}}(k)$ if and only if there exist $X\in \mathbb {S} ^{n}$ and $\epsilon \in \mathbb {R} _{>0}$, and $F\in \mathbb {R} ^{n\times n}$, where $X>0$, such that
${\begin{bmatrix}0&-kX&X&0\\*&0&0&-X\\*&*&0&-kX\\*&*&*&0\end{bmatrix}}+He{\begin{Bmatrix}{\begin{bmatrix}A&0\\1&0\\0&1\\0&A\end{bmatrix}}{\begin{bmatrix}F&0\\0&F\end{bmatrix}}{\begin{bmatrix}k1&-\epsilon k1&\epsilon 1&1\\-1&-\epsilon 1&\epsilon k1&k1\end{bmatrix}}\end{Bmatrix}}<0$.
Moreover, for every $X$ that satisfies
${\begin{bmatrix}k(AX+XA^{T})&AX-XA^{T}\\*&k(AX+XA^{T})\end{bmatrix}}<0$,
$X$ and $F=-\epsilon ^{-1}(A-\epsilon ^{-1}1)^{-1}X$ are solutions to
${\begin{bmatrix}0&-kX&X&0\\*&0&0&-X\\*&*&0&-kX\\*&*&*&0\end{bmatrix}}+He{\begin{Bmatrix}{\begin{bmatrix}A&0\\1&0\\0&1\\0&A\end{bmatrix}}{\begin{bmatrix}F&0\\0&F\end{bmatrix}}{\begin{bmatrix}k1&-\epsilon k1&\epsilon 1&1\\-1&-\epsilon 1&\epsilon k1&k1\end{bmatrix}}\end{Bmatrix}}<0$

## External Links

* LMI Methods in Optimal and Robust Control - A course on LMIs in Control by Matthew Peet.
* LMIs in Systems and Control Theory - A downloadable book on LMIs by Stephen Boyd.
* LMI Properties and Applications in Systems, Stability, and Control Theory - A downloadable book on LMIs by Ryan James Caverly and James Richard Forbes.
# LMIs in Control/Click here to continue/LMIs in system and stability Theory/Continuous- time Lyapunov Stability

LMIs in Control/Click here to continue/LMIs in system and stability Theory/Continuous- time Lyapunov Stability
The System:
    ${\begin{aligned}{\dot {x}}(t)&=Ax(t)\\x(0)&=x_{0}\end{aligned}}$
The Optimization Problem:
The LMI: The Lyapunov Inequality
       ${\begin{aligned}{\text{Find}}\;X:&\\{\begin{bmatrix}X\end{bmatrix}}&>0\\{\begin{bmatrix}A^{T}X+XA\end{bmatrix}}&<0\end{aligned}}$
Conclusion:
If feasible, System is internally exponentially Stable for any $x_{0}\in \mathbb {R} ^{n}$.
Related LMIs:
# LMIs in Control/Click here to continue/LMIs in system and stability Theory/Continuous-time Detectability
## Detectability LMI

Detectability is a weaker version of observability. A system is detectable if all unstable modes of the system are observable, whereas observability requires all modes to be observable. This implies that if a system is observable it will also be detectable. The LMI condition to determine detectability of the pair $(A,C)$ is shown below.

## The System

    ${\begin{aligned}{\dot {x}}(t)&=Ax(t)+Bu(t),\\y(t)&=Cx(t)+Du(t),\\x(0)&=x_{0},\end{aligned}}$
where $x(t)\in \mathbb {R} ^{n}$, $u(t)\in \mathbb {R} ^{m}$, at any $t\in \mathbb {R}$.

## The Data

The matrices necessary for this LMI are $A$ and $C$. There is no restriction on the stability of $A$.

## The LMI: Detectability LMI

$(A,B)$ is detectable if and only if there exists $X>0$ such that
       $AX+XA^{T}-B^{T}B<0$.

## Conclusion:

If we are able to find $X>0$ such that the above LMI holds it means the matrix pair $(A,C)$ is detectable. In words, a system pair $(A,C)$ is detectable if the unobservable states asymptotically approach the origin. This is a weaker condition than observability since observability requires that all initial states must be able to be uniquely determined in a finite time interval given knowledge of the input $u(t)$ and output $y(t)$.

## Implementation

This implementation requires Yalmip and Sedumi.
https://github.com/eoskowro/LMI/blob/master/Detectability_LMI.m

## Related LMIs

Stabilizability LMI
Hurwitz Stability LMI
Controllability Grammian LMI
Observability Grammian LMI

## External Links

A list of references documenting and validating the LMI.
* LMI Methods in Optimal and Robust Control - A course on LMIs in Control by Matthew Peet.
* LMI Properties and Applications in Systems, Stability, and Control Theory - A List of LMIs by Ryan Caverly and James Forbes.
* LMIs in Systems and Control Theory - A downloadable book on LMIs by Stephen Boyd.
* LMIs in Control Systems: Analysis, Design and Applications - by Guang-Ren Duan and Hai-Hua Yu, CRC Press, Taylor & amp; Francis Group, 2013, Section 6.1.1 and Table 6.1 pp. 166–170, 192.
* A Course in Robust Control Theory: a Convex Approach, - by Geir E. Dullerud and Fernando G. Paganini, Springer, 2011, Section 2.2.3, pp. 71-73.
# Lineage 2/The Player Character (PC) Classes & Character Guides/Warrior

Warriors are offensive melee warriors. They wear heavy or light armor, and wield swords, blunts, and polearms.

# Key Skills

Format: Name (max Level) - Description (Level [+Number of Advancements] [/ ...])

## Active

* Accuracy (1) - Increases accuracy. Continuously consumes MP. (24)
* Battle Roar (1) - Temporarily increases maximum HP and restores HP. (28)
* Detect Insect Weakness (1) - Temporarily increases P. Atk. against insects. (32)
* Lionheart (1) - Temporarily increases resistance to Paralysis, Hold, Sleep or Shock attack. (36)
* Power Smash (15) - Enhanced weapon swing. Usable when a sword or blunt weapon is equipped. Over-hit available. (20+3/24+3/28+3/32+3/36+3)
* Relax (1) - Recovers HP quickly when sitting. (5)
* Stun Attack (15) - Stunning attack that inflicts shock and damage upon the enemy. Used with a blunt weapon. Over-hit possible. (20+3/24+3/28+3/32+2/36+3)
* Vicious Stance (5) - Increases the degree of damage being given to an enemy during a critical attack. One's MP will be consumed continuously. (20/24/28/32/36)
* War Cry (1) - Momentarily increases P. Atk. (20)
* Wild Sweep (15) - Uses a pole-arm weapon to strike multiple enemies. Over-hit available. (20+3/24+3/28+3/32+3/32+3)

## Passive

* Heavy Armor Mastery (13) - Increases one's P. Def. when one is wearing heavy armor. (20+2/24+2/28+3/32+3/36+3)
* Light Armor Mastery (13) - Increases P. Def. and Evasion when wearing light armor. (20+2/24+2/28+3/32+3/36+3)
* Polearm Mastery (8) - Increases P. Atk. when using a polearm. (20/24/28+2/32+2/36+2)
* Sword Blunt Mastery (8) - Increases one's P. Atk when one is using a sword or blunt type of weapon. (20/24/28+2/32+2/36+2)
* Vital Force (2) - Allows quick recovery while sitting. (24/32)

## Outdated

* Armor Mastery (5) - Increase p.def. (5/10+2/15+2)
* Moral Blow (9) - Potentially deadly attack upon the enemy. Dagger only. (5+3/10+3/15+3)
* Power Shot (9) - Causes great damage from afar when using a bow. Over-hit available. (5+3/10+3/15+3)
* Power Strike (9) - Gathers power for a fierce strike. Sword/blunt only. Over-hit available. (5+3/10+3/15+3)
* Weapon Mastery (3) - Increases p.atk (5/10/15)
# Lineage 2/The Player Character (PC) Classes & Character Guides/Temple Knight

The Temple Knight is the Elven defensive warrior. They use swords and blunts in conjunction with a shield and heavy armor. They get a few base skills targetted for foiling the undead, as well as some ability of summoning cubics.
Like all pure tank classes, the Temple Knight has Aggression and Hate Aura to attract mobs to focus their attacks on them (instead of on other party members), and they have Ultimate Defense to increase their defense values, and Deflect Arrow to make bow attacks much less efficient. They also has the passive skills Final Fortress to increase p.def with falling hp, and Shield Mastery to improve the effectivity of shields.
Like all Elven characters, the Temple Knight has the ability to heal themselves using Elemental Heal, and the summoned Healing Cubic.
# Lineage 2/The Player Character (PC) Classes & Character Guides/Warlord

The Warlord is a melee warrior class. They use heavy and light armor and specialize into polearms.

# Key Skills

Format: Name (max Level) - Description (Level [+Number of Advancements] [/ ...])

## Selfbuffs

* Accuracy (1) - Increases accuracy. Continuously consumes MP. (24)
* Battle Roar (6) - Temporarily increases maximum HP and restores HP. (28/40/49/58/64/70)
* Detect Animal Weakness (1) - Temporarily increases P. Atk. against animals. (40)
* Detect Dragon Weakness (1) - Temporarily increases P. Atk. against dragons. (58)
* Detect Insect Weakness (1) - Temporarily increases P. Atk. against insects. (32)
* Detect Monster Weakness (1) - Temporarily increases P. Atk. against monsters. (52)
* Detect Plant Weakness (1) - Temporarily increases P. Atk. against plants. (46)
* Lionheart (3) - Temporarily increases resistance to Paralysis, Hold, Sleep or Shock attack. (36/49/62)
* Relax (1) - Recovers HP quickly when sitting. (5)
* Revival (1) - PC fully recovers. Can only be used 1/10 of maximum HP. (55)
* Thrill Fight (2) - Temporarily decreases Speed and increases Atk. Spd. (46/55)
* War Cry (1) - Momentarily increases P. Atk. (20)

## Attacks

* Thunder Storm (37) - Stun and damage area attack. Over-hit available. Polearms only. (40+3/43+3/46+3/49+3/52+3/55+3/58+3/60+2/62+2/64+2/66+2/68+2/70+2/72+2/74+2)
* Vicious Stance (20) - Empowers critical damage. Continuously consumes MP. (20/24/28/32/36/40/43/46/49/52/55/58/60/62/64/66/68/70/72/74)
* Whirlwind (37) - Attack all surrounding enemies. Over-hit available. Polearms only. (40+3/43+3/46+3/49+3/52+3/55+3/58+3/60+2/62+2/64+2/66+2/68+2/70+2/72+2/74+2)
* Wild Sweep (15) - Uses a pole-arm weapon to strike multiple enemies. Over-hit available. (20+3/24+3/28+3/32+3/36+3)

## Other Active

* Focus Attack (1) - Attacks one enemy at a time. Increases Accuracy. Polearm only. (40)
* Howl (14) - Instantly reduces P. Atk. of surrounding enemies. (43/46/49/52/55/58/60/62/64/66/68/70/72/74)
* Provoke (3) - Draws in monsters towards oneself from a large surrounding area. (43/55/60)
* Wrath (10) - Decreases CP of nearby enemies by a fixed percent. Polearms only. (66+2/68+2/70+2/72+2/74+2)

## Passive

* Final Frenzy (14) - Increases P.Atk for falling hp. (43/46/49/52/55/58/60/62/64/66/68/70/72/74)
* Heavy Armor Mastery (50) - Increases P. Def. Heavy armor only. (20+2/24+2/28+3/32+3/36+3/40+3/43+3/46+3/49+3/52+3/55+3/58+3/60+2/62+2/64+2/66+2/68+2/70+2/72+2/74+2)
* Light Armor Mastery (50) - Increases P. Def. and Evasion when wearing light armor. (20+2/24+2/28+3/32+3/36+3/40+3/43+3/46+3/49+3/52+3/55+3/58+3/60+2/62+2/64+2/66+2/68+2/70+2/72+2/74+2)
* Polearm Mastery (45) - Increases P. Atk. when using a polearm. (20/24/28+2/32+2/36+2/40+3/43+3/46+3/49+3/52+3/55+3/58+3/60+2/62+2/64+2/66+2/68+2/70+2/72+2/74+2)
* Vital Force (2) - Allows quick recovery while sitting. (24/32)
# Linear Algebra/Addition, Multiplication, and Transpose
## Addition and subtraction

Two matrices can only be added or subtracted if they have the same size. Matrix addition and subtraction are done entry-wise, which means that each entry in A+B is the sum of the corresponding entries in A and B.
    $A={\begin{bmatrix}7&&5&&3\\4&&0&&5\end{bmatrix}}\qquad B={\begin{bmatrix}1&&1&&1\\-1&&3&&2\end{bmatrix}}$
Here is an example of matrix addition
    $A+B={\begin{bmatrix}7+1&&5+1&&3+1\\4-1&&0+3&&5+2\end{bmatrix}}={\begin{bmatrix}8&&6&&4\\3&&3&&7\end{bmatrix}}$
And an example of subtraction
    $A-B={\begin{bmatrix}7-1&&5-1&&3-1\\4+1&&0-3&&5-2\end{bmatrix}}={\begin{bmatrix}6&&4&&2\\5&&-3&&3\end{bmatrix}}$
Remember you can not add or subtract two matrices of different sizes.
The following rules applies to sums and scalar multiples of matrices. Let $A,B,C$ be matrices of the same size, and let $r,s$ be scalars.
* $A+B=B+A$
* $(A+B)+C=A+(B+C)$
* $A+0=A$
* $r(A+B)=rA+rB$
* $(r+s)A=rA+sA$
* $r(sA)=(rs)A$

## Multiplication

What is matrix multiplication? You can multiply two matrices if, and only if, the number of columns in the first matrix equals the number of rows in the second matrix.
Otherwise, the product of two matrices is undefined. The product matrix's dimensions are
    $\to ({\text{rows of first matrix}})\times ({\text{columns of the second matrix}})$
In above multiplication, the matrices cannot be multiplied since the number of columns in the 1st one, matrix $A$ is not equals the number of rows in the 2nd, matrix $B$ . The Dimensions of the product matrix. Rows of 1st matrix × Columns of 2nd

## Powers

If $A$ is an $n\times n$ matrix and if $k$ is a positive integer, then $A^{k}$ denotes the product of $k$ copies of $A$
    $A^{k}={\begin{matrix}\underbrace {A\cdots A} \\k\end{matrix}}$
If $A$ is non-zero and if ${\rm {x}}$ is in $\mathbb {R} ^{n}$ , then $A^{k}{\rm {x}}$ is the result of left-multiplying ${\rm {x}}$ by $A$ repeatedly $k$ times. If $k=0$ , then $A^{0}{\rm {x}}$ should be ${\rm {x}}$ itself. Thus $A^{0}$ is interpreted as the identity matrix.

## Transpose

Given the $m\times n$ matrix $A$ , the transpose of $A$ is the $n\times m$ , denoted $A^{T}$ , whose columns are formed from the corresponding rows of $A$ .
For example
    $A={\begin{bmatrix}a&&b\\c&&d\end{bmatrix}}\qquad B={\begin{bmatrix}3&&5\\2&&7\\6&&9\\1&&0\\5&&2\end{bmatrix}}$
    $A^{T}={\begin{bmatrix}a&&c\\b&&d\end{bmatrix}}\qquad B^{T}={\begin{bmatrix}3&&2&&6&&1&&5\\5&&7&&9&&0&&2\end{bmatrix}}$
The following rules applied when working with transposing
* $(A^{T})^{T}=A$
* $(A+B)^{T}=A^{T}+B^{T}$
* For any scalar $r$ , $(rA)^{T}=rA^{T}$
* $(AB)^{T}=B^{T}A^{T}$
The 4th rule can be generalize to products of more than two factors, as "The transpose of a product of matrices equals the product of their transposes in the reverse order." Meaning
    $(a_{1},a_{2},a_{3},\dots ,a_{n})^{T}=a_{n}^{T},\dots ,a_{3}^{T},a_{2}^{T},a_{1}^{T}$
# Minecraft/Skeleton
## Skeleton

Minecraft
Blocks | Items | Mobs
Structures
Miscellaneous
Guides
Skeletons are undead mobs that carry a bow and arrows, and are bent on your destruction.

## Abilities and strengths/weaknesses

Undead
This mob is undead, and takes additional damage from weapons with the smite enchantment. This mob is healed by potions of harming and damaged by potions of healing.
Sunlight: burn
This mob burns in sunlight unless it is wearing a helmet or has the "Fire Risistance" buff.

### Drops

* 1 - 2 bones
* 0 - 3 arrows
* bow (rare)
* 5 exp. points (+2 for equipment)
* skeleton skull if killed by a charged creeper

## Description

These horrors carry bows and shoot at you. They are attacked by wolves, and will run from them. As they run, they will try to shoot at it.

## Strategies

Their bow makes it hard to hit. Try circling it, blocking incoming attacks with a shield or shooting at it from a distance with a bow and arrows or a trident.

## Varients

The stray spawn in frozen biomes if there is exposed sky.
wither skeletons are Nether varients.

## Achievements
### Sniper duel

Kill a skeleton from over fifty blocks away.

## Mobs

* Animals
       Sheep
       Pig
       Cow
       Chicken
       Bat
       Cat
       Ocelot
       Wolf
       Fox
       Horse
       Donkey
       Mule
       Llama
       Bunny
       Frog
       Axolotl
       Tadpole
       Panda
       Mooshroom
* Monsters
       Creeper
       Zombie
       Skeleton
       Stray
       Witch
       Slime
       Phantom
       Warden
* Ocean
       Cod
       Pufferfish
       Salmon
       Tropical Fish
       Squid
       Glow Squid
       Dolphin
       Turtle
       Guardian
       Elder Guardian
* Nether
       Blaze
       Ghast
       Piglin
       Zombified Piglin
       Hoglin
       Strider
       Wither Skeleton
* End
       Enderman
       Endermite
       Shulker
* Illager/Villagers
       Villager
       Wandering Trader
       Illager
       Pillager
       Evoker
       Vex
       Ravanger
* Utility
       Snow Golem
       Iron Golem
       Allay
* Bosses
       Wither
       Ender Dragon
* Secret mobs
       Npc