Matrix transpose and properties — Hinglish notes

1.1.7 · AI-ML › Linear Algebra Essentials

Transpose HAI kya?

Yeh definition KYU? Main diagonal (entries $A_{ii}$ ) apni jagah rahi — ek "mirror line" ki tarah. Off-diagonal entries apne partners se swap karti hain: jo $(j, i)$ pe tha woh $(i, j)$ pe aa jaata hai. Isliye shape $m \times n$ se $n \times m$ ho jaata hai.

Figure — Matrix transpose and properties

Core properties KAISE derive hoti hain

Neeche sab kuch ek hi rule $(A^{⊤})_{ij} = A_{j i}$ se prove kiya gaya hai. Kuch bhi raatna nahi hai.

1. Transpose self-inverse hai

$((A^{⊤})^{⊤})_{ij} = (A^{⊤})_{j i} = A_{ij} ⟹ (A^{⊤})^{⊤} = A .$ Kyun? Do baar flip karne se har entry apne ghar wapas aa jaati hai. Isliye $⊤$ ek involution hai.

2. Linearity: $(A + B)^{⊤} = A^{⊤} + B^{⊤}$ aur $(c A)^{⊤} = c A^{⊤}$

$((A + B)^{⊤})_{ij} = (A + B)_{j i} = A_{j i} + B_{j i} = (A^{⊤})_{ij} + (B^{⊤})_{ij} .$ Kyun? Transpose sirf entries ko move karta hai; addition entry-by-entry hoti hai, isliye dono operations commute karti hain.

3. Reverse-order rule: $(A B)^{⊤} = B^{⊤} A^{⊤}$

Matrix-product definition $(A B)_{ij} = \sum_{k} A_{ik} B_{k j}$ se shuru karte hain: $((A B)^{⊤})_{ij} = (A B)_{j i} = \sum_{k} A_{j k} B_{k i} .$ Ab transpose rule se har factor ko rewrite karo ( $A_{j k} = (A^{⊤})_{k j}$ , $B_{k i} = (B^{⊤})_{ik}$ ): $= \sum_{k} (B^{⊤})_{ik} (A^{⊤})_{k j} = (B^{⊤} A^{⊤})_{ij} .$ Isliye $(A B)^{⊤} = B^{⊤} A^{⊤}$ .

4. Inverse ka transpose: $(A^{- 1})^{⊤} = (A^{⊤})^{- 1}$

$A A^{- 1} = I$ ke dono sides ko rule 3 se transpose karo (aur $I^{⊤} = I$ ): $(A^{- 1})^{⊤} A^{⊤} = I ⟹ (A^{- 1})^{⊤} = (A^{⊤})^{- 1} .$ Notation $A^{- ⊤}$ is common object ka shorthand hai.

5. Determinant unchanged rehta hai: $det (A^{⊤}) = det (A)$

Leibniz formula permutations $σ$ pe sum karta hai: transpose ke under har product $\prod_{i} A_{i, σ (i)}$ , $\prod_{i} A_{σ (i), i}$ ban jaata hai, jo $σ^{- 1}$ se reindex kiya hua same sum hai (same sign ke saath). Toh total identical hai.

Symmetric & skew-symmetric matrices

Transpose hi dot-product mover hai

Yeh bhi: vector $x$ ke liye, $x^{⊤} x = \sum_{i} x_{i}^{2} = ∥ x ∥^{2}$ . Transpose ek column ko woh machine bana deta hai jo squared length compute karta hai.

Worked examples

Common mistakes (steel-manned)

Recall Feynman: 12-saal ke bacche ko samjhao

Ek class ki seating chart socho jo grid ki tarah bani hai. Har number kehta hai "yeh baccha row $i$ , seat $j$ mein hai." Transpose woh hota hai jo tumhe milta hai agar tum grid ko iske diagonal mirror ke saath side mein ghuma do — jaise koi photo ko 90° tilt karo. Jo baccha "row 3, seat 1" mein tha woh ab "row 1, seat 3" mein hai. Tumne koi baccha add ya remove nahi kiya, bas grid ko padhne ka tarika badla. Aur ek cool rule: agar pehle task $A$ karo phir task $B$ aur poori cheez flip karo, toh yeh same hai jaise $B$ flip karo, phir $A$ flip karo — tumhe unhe reverse order mein undo karna hoga, jaise shoes utaarne se pehle moje utaaro.

Flashcards

Transpose ke liye defining entry rule kya hai?

(A^{⊤})_{ij} = A_{j i}

— dono indices swap karo.

Transpose ek

m \times n

matrix ki shape kaise change karta hai?

Yeh

n \times m

ho jaati hai.

Reverse-order (socks-and-shoes) rule batao.

(A B)^{⊤} = B^{⊤} A^{⊤}

Derive karo kyun

(A B)^{⊤} = B^{⊤} A^{⊤}

hota hai na ki

A^{⊤} B^{⊤}

(A B)_{ij}^{⊤} = \sum_{k} A_{j k} B_{k i} = \sum_{k} (B^{⊤})_{ik} (A^{⊤})_{k j}

; shapes bhi reversal force karti hain.

(A^{⊤})^{⊤}

kya hai?

A

— transpose ek involution (self-inverse) hai.

A

symmetric kab hota hai? Skew-symmetric kab?

Symmetric:

A^{⊤} = A

. Skew:

A^{⊤} = - A

(zero diagonal).

A^{⊤} A

hamesha symmetric kyun hota hai?

(A^{⊤} A)^{⊤} = A^{⊤} (A^{⊤})^{⊤} = A^{⊤} A

Kisi bhi square

A

ko symmetric + skew parts mein kaise split karoge?

A = \frac{1}{2} (A + A^{⊤}) + \frac{1}{2} (A - A^{⊤})

det (A)

aur

det (A^{⊤})

mein kya relation hai?

Dono equal hain; isliye same characteristic polynomial aur eigenvalues hote hain.

(A^{- 1})^{⊤}

kya hota hai?

(A^{⊤})^{- 1}

(

A A^{- 1} = I

ko transpose karke prove hota hai).

Transpose ki inner-product / adjoint property kya hai?

⟨ A x, y ⟩ = ⟨ x, A^{⊤} y ⟩

Skew-symmetric matrix ka diagonal zero kyun hona zaroori hai?

A_{ii} = - A_{ii} \Rightarrow A_{ii} = 0

Connections

Matrix multiplication — reverse-order rule product definition pe depend karta hai.
Symmetric matrices and the Spectral Theorem — real eigenvalues, orthogonal eigenvectors.
Dot product and inner products — transpose = adjoint jo $A$ ko $⟨ \cdot, \cdot ⟩$ ke across move karta hai.
Covariance and Gram matrices — $A^{⊤} A$ symmetric; PCA mein use hota hai.
Normal equations and least squares — $A^{⊤} A x = A^{⊤} b$ .
Determinant aur Matrix inverse — invariance aur $(A^{- 1})^{⊤}$ .
Orthogonal matrices — $Q^{⊤} Q = I$ se define hote hain.