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# Tensor Shape

### 1. Mathematical Definition

The shape of a tensor is an ordered tuple (n₁, n₂, ..., nₖ) where each nᵢ represents the size of the i-th dimension. It describes the number of elements along each dimension of the tensor.

### 2. Mathematical Examples by Dimensionality

#### 2.1 Scalar (Shape: ())

Shape: ()

Example: 5

Components: Single value

#### 2.2 Vector (Shape: (n))

Shape: (3)

Example: $$v = \begin{bmatrix} 1 \ 2 \ 3 \end{bmatrix}$$

Basis vectors in ℝ³: $$e\_1 = \begin{bmatrix} 1 \ 0 \ 0 \end{bmatrix}, \quad e\_2 = \begin{bmatrix} 0 \ 1 \ 0 \end{bmatrix}, \quad e\_3 = \begin{bmatrix} 0 \ 0 \ 1 \end{bmatrix}$$

#### 2.3 Matrix (Shape: (m,n))

Shape: (2,3)

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

Shape: (3,3)

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

#### 2.4 3D Tensor (Shape: (l,m,n))

Shape: (2,2,3)

Example: $$T = \begin{bmatrix} \begin{bmatrix} 1 & 2 & 3 \ 4 & 5 & 6 \end{bmatrix}, \begin{bmatrix} 7 & 8 & 9 \ 10 & 11 & 12 \end{bmatrix} \end{bmatrix}$$

RGB Image Example (Shape: (height, width,3)):

Shape: (2,3,3)

$$T = \begin{bmatrix} \begin{bmatrix} 255 & 0 & 0 \ 0 & 255 & 0 \ 0 & 0 & 255 \end{bmatrix}, \begin{bmatrix} 128 & 0 & 0 \ 0 & 128 & 0 \ 0 & 0 & 128 \end{bmatrix} \end{bmatrix}$$

### 3. Mathematical Operations with Shapes

#### 3.1 Reshaping Operations

Original shape: (4,) → vector $$\begin{bmatrix} 1 \ 2 \ 3 \ 4 \end{bmatrix}$$

Reshaped to (2,3) → matrix $$\begin{bmatrix} 1 & 2 & 3 \ 4 & 5 & 6 \end{bmatrix}$$

Reshaped to (3,2) → matrix $$\begin{bmatrix} 1 & 2 \ 3 & 4 \ 5 & 6 \end{bmatrix}$$

#### 3.2 Shape Compatibility in Operations

Matrix Multiplication ($$A×B$$):

A shape: $$(m,n)$$

B shape: $$(n,p)$$

Result shape: $$(m,p)$$

Example: $$(2,3) \times (3,2) \rightarrow (2,2)$$

$$\begin{bmatrix} 1 & 2 & 3 \ 4 & 5 & 6 \end{bmatrix} \times \begin{bmatrix} 7 & 8 \ 9 & 10 \ 11 & 12 \end{bmatrix} \begin{bmatrix} 58 & 64 \ 139 & 154 \end{bmatrix}$$

#### 3.3 Broadcasting Rules

Vector + Scalar: Shape (3,) + () → (3,)

$$\begin{bmatrix} 1 \ 2 \ 3 \end{bmatrix} + 4 = \begin{bmatrix} 4 \ 6 \ 7 \end{bmatrix}$$

Matrix + Vector: Shape (2,3) + (3,) → (2,3)

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

### 4. Special Tensor Shapes in Mathematics

#### 4.1 Square Matrices (n,n)

Shape: (2,2)

$$\begin{bmatrix} a & b \ c & d \end{bmatrix}$$

Shape: (3,3)

$$\begin{bmatrix} a & b & c \ d & e & f \ g & h & i \end{bmatrix}$$

#### 4.2 Diagonal Matrices

Shape: (3,3)

$$\begin{bmatrix} \lambda\_1 & 0 & 0 \ 0 & \lambda\_2 & 0 \ 0 & 0 & \lambda\_3 \end{bmatrix}$$

#### 4.3 Symmetric Tensors

This shows a symmetric matrix where the entries satisfy $$b\_{ij} = b\_{ji}$$ .

Shape: (3,3)

$$\begin{bmatrix} a & b & c \ b & d & e \ c & e & f \end{bmatrix}$$

### 5. Applications with Specific Shapes

#### 5.1 Physics Examples

Stress Tensor (Shape: (3,3)):

$$\begin{bmatrix} \sigma\_{xx} & \sigma\_{xy} & \sigma\_{xz} \ \sigma\_{yx} & \sigma\_{yy} & \sigma\_{yz} \ \sigma\_{zx} & \sigma\_{zy} & \sigma\_{zz} \end{bmatrix}$$

Electromagnetic Field Tensor (Shape: (4,4)):

$$\begin{bmatrix} 0 & -E\_x & -E\_y & -E\_z \ E\_x & 0 & -B\_z & B\_y \ E\_y & B\_z & 0 & -B\_x \ E\_z & -B\_y & B\_x & 0 \end{bmatrix}$$

#### 5.2 Linear Algebra Examples

```mathematica
yRotation Matrix (Shape: (3,3)):
[cos θ  -sin θ  0]
[sin θ   cos θ  0]
[0       0      1]

Projection Matrix (Shape: (n,n)):
P = A(AᵀA)⁻¹Aᵀ
```

#### 5.3 Data Science Examples

```mathematica
Feature Matrix (Shape: (samples, features)):
Shape: (3,4)
[x₁₁ x₁₂ x₁₃ x₁₄]
[x₂₁ x₂₂ x₂₃ x₂₄]
[x₃₁ x₃₂ x₃₃ x₃₄]

Neural Network Layer (Shape: (batch_size, neurons)):
Shape: (2,3)
[n₁₁ n₁₂ n₁₃]
[n₂₁ n₂₂ n₂₃]
```

### 6. Shape Transformations

#### 6.1 Transpose Operation

Original Shape: (2,3)

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

Transposed Shape: (3,2)

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

#### 6.2 Flatten Operation

Original Shape: (2,2,2)

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

Flattened Shape: (8,)

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