1.2 Matritsa va Chiziqli Algebra — Nazariya

Hafta: 3 | Masalalar: 30 | Qiyinlik: ⭐⭐⭐

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Kirish

Matritsa — sonlarni to'rtburchak jadval ko'rinishida tartibga solish usuli. Robotika va dronlarda matritsalar hamma joyda:

• 🔄 Koordinata transformatsiyalari — robot qo'lini boshqarish
• 🎯 Aylanishlar — dronning yo'nalishini o'zgartirish
• 📊 Tenglamalar sistemasi — kinematika yechimlari
• 🤖 Kalman filtri — sensor ma'lumotlarini qayta ishlash

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1. Matritsa Asoslari

Ta'rif

$m \times n$ matritsa — $m$ ta qator va $n$ ta ustundan iborat jadval:

$$A = \begin{bmatrix} a_{11} & a_{12} & \cdots & a_{1n} \\ a_{21} & a_{22} & \cdots & a_{2n} \\ \vdots & \vdots & \ddots & \vdots \\ a_{m1} & a_{m2} & \cdots & a_{mn} \end{bmatrix}$$

Maxsus Matritsalar

Kvadrat matritsa ($n \times n$):

$$A = \begin{bmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \\ 7 & 8 & 9 \end{bmatrix}$$

Birlik matritsa (Identity):

$$I = \begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{bmatrix}$$

Diagonal matritsa:

$$D = \begin{bmatrix} d_1 & 0 & 0 \\ 0 & d_2 & 0 \\ 0 & 0 & d_3 \end{bmatrix}$$

Nol matritsa:

$$O = \begin{bmatrix} 0 & 0 \\ 0 & 0 \end{bmatrix}$$

Simmetrik matritsa ($A = A^T$):

$$A = \begin{bmatrix} 1 & 2 & 3 \\ 2 & 4 & 5 \\ 3 & 5 & 6 \end{bmatrix}$$

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2. Matritsa Amallari

Qo'shish va Ayirish

Faqat bir xil o'lchamdagi matritsalar uchun:

$$A + B = \begin{bmatrix} a_{11}+b_{11} & a_{12}+b_{12} \\ a_{21}+b_{21} & a_{22}+b_{22} \end{bmatrix}$$

Skalyarga Ko'paytirish

$$kA = \begin{bmatrix} ka_{11} & ka_{12} \\ ka_{21} & ka_{22} \end{bmatrix}$$

Matritsa Ko'paytirish

$A$ ($m \times n$) va $B$ ($n \times p$) uchun natija $C$ ($m \times p$):

$$C_{ij} = \sum_{k=1}^{n} a_{ik} \cdot b_{kj}$$

Muhim

Matritsa ko'paytirish kommutativ emas: $AB \neq BA$

Misol

$$\begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix} \times \begin{bmatrix} 5 & 6 \\ 7 & 8 \end{bmatrix} = \begin{bmatrix} 1(5)+2(7) & 1(6)+2(8) \\ 3(5)+4(7) & 3(6)+4(8) \end{bmatrix} = \begin{bmatrix} 19 & 22 \\ 43 & 50 \end{bmatrix}$$

Transpozitsiya

Qator va ustunlarni almashtirish:

$$A = \begin{bmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \end{bmatrix} \Rightarrow A^T = \begin{bmatrix} 1 & 4 \\ 2 & 5 \\ 3 & 6 \end{bmatrix}$$

Xossalari:

• $(A^T)^T = A$
• $(A + B)^T = A^T + B^T$
• $(AB)^T = B^T A^T$
• $(kA)^T = kA^T$

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3. Determinant

2×2 Matritsa

$$\det(A) = |A| = \begin{vmatrix} a & b \\ c & d \end{vmatrix} = ad - bc$$

3×3 Matritsa (Sarrus qoidasi)

$$\det(A) = \begin{vmatrix} a & b & c \\ d & e & f \\ g & h & i \end{vmatrix} = aei + bfg + cdh - ceg - bdi - afh$$

Kofaktor usuli

$$\det(A) = \sum_{j=1}^{n} (-1)^{1+j} a_{1j} M_{1j}$$

Bu yerda $M_{1j}$ — minor (1-qator va j-ustunni olib tashlagandan keyingi determinant).

Xossalari

1. $\det(I) = 1$
2. $\det(A^T) = \det(A)$
3. $\det(AB) = \det(A) \cdot \det(B)$
4. $\det(kA) = k^n \det(A)$ ($n \times n$ matritsa uchun)
5. Agar qator/ustun nollardan iborat bo'lsa: $\det(A) = 0$

Geometrik ma'no

• 2D: parallellogramm yuzi
• 3D: parallelepiped hajmi
• $\det(A) = 0$ — tizim yechimga ega emas yoki cheksiz yechim

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4. Teskari Matritsa

Ta'rif

$A^{-1}$ — $A$ ning teskari matritsasi, agar:

$$AA^{-1} = A^{-1}A = I$$

Mavjudlik sharti

Teskari matritsa mavjud ⟺ $\det(A) \neq 0$

2×2 Matritsa uchun

$$A^{-1} = \frac{1}{\det(A)} \begin{bmatrix} d & -b \\ -c & a \end{bmatrix}$$

Misol

$$A = \begin{bmatrix} 4 & 7 \\ 2 & 6 \end{bmatrix}$$

$\det(A) = 4(6) - 7(2) = 24 - 14 = 10$

$$A^{-1} = \frac{1}{10} \begin{bmatrix} 6 & -7 \\ -2 & 4 \end{bmatrix} = \begin{bmatrix} 0.6 & -0.7 \\ -0.2 & 0.4 \end{bmatrix}$$

Umumiy usul (Adjoint)

$$A^{-1} = \frac{1}{\det(A)} \text{adj}(A)$$

Bu yerda $\text{adj}(A)$ — kofaktorlar matritsasining transpozitsiyasi.

Xossalari

1. $(A^{-1})^{-1} = A$
2. $(AB)^{-1} = B^{-1}A^{-1}$
3. $(A^T)^{-1} = (A^{-1})^T$
4. $\det(A^{-1}) = \frac{1}{\det(A)}$

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5. Chiziqli Tenglamalar Sistemasi

Matritsa ko'rinishi

$$\begin{cases} a_{11}x_1 + a_{12}x_2 + \cdots + a_{1n}x_n = b_1 \\ a_{21}x_1 + a_{22}x_2 + \cdots + a_{2n}x_n = b_2 \\ \vdots \\ a_{m1}x_1 + a_{m2}x_2 + \cdots + a_{mn}x_n = b_m \end{cases}$$

Matritsa shaklida: $A\vec{x} = \vec{b}$

Yechish usullari

1. Teskari matritsa:

$$\vec{x} = A^{-1}\vec{b}$$

2. Kramer qoidasi:

$$x_i = \frac{\det(A_i)}{\det(A)}$$

Bu yerda $A_i$ — $A$ ning $i$-ustuni $\vec{b}$ bilan almashtirilgan matritsa.

3. Gauss eliminatsiyasi: Kengaytirilgan matritsani qatorli operatsiyalar bilan soddalashtirish.

Gauss Usuli

$$[A|\vec{b}] \rightarrow [I|\vec{x}]$$

Qatorli operatsiyalar:

1. Ikki qatorni almashtirish
2. Qatorni nolmas songa ko'paytirish
3. Bir qatorga boshqa qatorning ko'paytmasini qo'shish

Misol

$$\begin{cases} 2x + y = 5 \\ x - y = 1 \end{cases}$$$$\begin{bmatrix} 2 & 1 & | & 5 \\ 1 & -1 & | & 1 \end{bmatrix} \xrightarrow{R_1 \leftrightarrow R_2} \begin{bmatrix} 1 & -1 & | & 1 \\ 2 & 1 & | & 5 \end{bmatrix}$$$$\xrightarrow{R_2 - 2R_1} \begin{bmatrix} 1 & -1 & | & 1 \\ 0 & 3 & | & 3 \end{bmatrix} \xrightarrow{R_2/3} \begin{bmatrix} 1 & -1 & | & 1 \\ 0 & 1 & | & 1 \end{bmatrix}$$$$\xrightarrow{R_1 + R_2} \begin{bmatrix} 1 & 0 & | & 2 \\ 0 & 1 & | & 1 \end{bmatrix}$$

Javob: $x = 2$, $y = 1$

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6. Xos Qiymatlar va Xos Vektorlar

Ta'rif

$\lambda$ — $A$ ning xos qiymati (eigenvalue), agar:

$$A\vec{v} = \lambda\vec{v}$$

Bu yerda $\vec{v} \neq \vec{0}$ — xos vektor (eigenvector).

Topish usuli

1. Xarakteristik tenglama:

$$\det(A - \lambda I) = 0$$

2. Xos vektorlar: Har bir $\lambda$ uchun $(A - \lambda I)\vec{v} = \vec{0}$ ni yeching.

Misol

$$A = \begin{bmatrix} 4 & 1 \\ 2 & 3 \end{bmatrix}$$

Xarakteristik tenglama:

$$\det\begin{bmatrix} 4-\lambda & 1 \\ 2 & 3-\lambda \end{bmatrix} = (4-\lambda)(3-\lambda) - 2 = 0$$$$\lambda^2 - 7\lambda + 10 = 0 \Rightarrow \lambda_1 = 5, \lambda_2 = 2$$

$\lambda_1 = 5$ uchun xos vektor:

$$\begin{bmatrix} -1 & 1 \\ 2 & -2 \end{bmatrix}\vec{v} = 0 \Rightarrow \vec{v}_1 = \begin{bmatrix} 1 \\ 1 \end{bmatrix}$$

$\lambda_2 = 2$ uchun:

$$\vec{v}_2 = \begin{bmatrix} 1 \\ -2 \end{bmatrix}$$

Qo'llanilish

• Barqarorlik tahlili — boshqarish tizimlari
• Tebranish modlari — mexanik tizimlar
• PCA — ma'lumotlarni tahlil qilish
• Inertia tensori — qattiq jism dinamikasi

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7. Matritsa Dekompozitsiyalari

LU Dekompozitsiya

$$A = LU$$

• $L$ — pastki uchburchak matritsa
• $U$ — yuqori uchburchak matritsa

Tenglamalar yechishda samarali: $A\vec{x} = \vec{b}$ → $L\vec{y} = \vec{b}$, $U\vec{x} = \vec{y}$

QR Dekompozitsiya

$$A = QR$$

• $Q$ — ortogonal matritsa ($Q^TQ = I$)
• $R$ — yuqori uchburchak matritsa

Xos qiymatlarni hisoblash va eng kichik kvadratlar usulida qo'llaniladi.

SVD (Singular Value Decomposition)

$$A = U\Sigma V^T$$

• $U$, $V$ — ortogonal matritsalar
• $\Sigma$ — diagonal matritsa (singular qiymatlar)

Qo'llanilish:

• Tasvirni siqish
• Pseudoinverse hisoblash
• Rang aniqlash

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8. Robotikada Qo'llanilish

Aylanish Matritsalari (2D)

$\theta$ burchakka aylanish:

$$R(\theta) = \begin{bmatrix} \cos\theta & -\sin\theta \\ \sin\theta & \cos\theta \end{bmatrix}$$

Aylanish Matritsalari (3D)

X o'qi atrofida:

$$R_x(\theta) = \begin{bmatrix} 1 & 0 & 0 \\ 0 & \cos\theta & -\sin\theta \\ 0 & \sin\theta & \cos\theta \end{bmatrix}$$

Y o'qi atrofida:

$$R_y(\theta) = \begin{bmatrix} \cos\theta & 0 & \sin\theta \\ 0 & 1 & 0 \\ -\sin\theta & 0 & \cos\theta \end{bmatrix}$$

Z o'qi atrofida:

$$R_z(\theta) = \begin{bmatrix} \cos\theta & -\sin\theta & 0 \\ \sin\theta & \cos\theta & 0 \\ 0 & 0 & 1 \end{bmatrix}$$

Homogen Transformatsiya (4×4)

Aylanish + siljish bir matritsada:

$$T = \begin{bmatrix} R & \vec{t} \\ 0 & 1 \end{bmatrix} = \begin{bmatrix} r_{11} & r_{12} & r_{13} & t_x \\ r_{21} & r_{22} & r_{23} & t_y \\ r_{31} & r_{32} & r_{33} & t_z \\ 0 & 0 & 0 & 1 \end{bmatrix}$$

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Xulosa

Tushuncha	Qo'llanilish
Determinant	Yechim mavjudligi, hajm
Teskari matritsa	Tenglamalar yechish
Xos qiymatlar	Barqarorlik, tebranish
Aylanish matritsasi	Robot/dron yo'nalishi
Homogen transformatsiya	Robot kinematikasi

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