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Matrix is a two dimensional arrangement of numbers.

Linear Algebra - Matrices

A matrix can be conceptualized as a vertical arrangement of its row vectors. If each row of a matrix $$H$$ is represented by a row vector $$h_i$$, then the entire matrix $$H$$ can be written as a stack of these vectors from $$h_0$$ to $$h_m$$. This is expressed as: $$H = \begin{bmatrix} h_0 \\ \vdots \\ h_m \end{bmatrix}$$

Matrix as a Stack of Row Vectors

The state of a model at a specific layer, often denoted by a capital letter like $$\mathbf{H}^l$$, is represented as an ordered sequence of vectors. For example, $$\mathbf{H}^l = \mathbf{h}_0^l \mathbf{h}_1^l \dots \mathbf{h}_m^l$$. Each bolded element $$\mathbf{h}_i^l$$ is a vector corresponding to the `i`-th position in the sequence at layer `l`. This collection of vectors encapsulates the full set of representations for that layer.

Representation of a Layer's State as a Sequence of Vectors

A language model processes the sentence 'The cat sat' and represents it internally as the following 3x4 matrix, where each row corresponds to a word in the sequence:

```
[[ 0.1, -0.4,  0.8,  0.2],
 [ 0.9,  0.2, -0.1, -0.5],
 [-0.3,  0.7,  0.6,  0.1]]
```

Analyze this matrix. Identify what the second row represents and explain the significance of the four numerical values within that specific row.

Interpreting Matrix Representations in a Language Model

A computational model processes the three-word sentence 'The cat sat'. After passing through one of the model's layers, the state is represented by the following 3x4 matrix. Analyze this matrix: What does each row likely represent? What do the dimensions (number of rows and columns) signify in this context?

Analyzing a Layer's State Matrix in a Language Model

A diagonal matrix is a matrix in which all entries outside the main diagonal are zero. Formally, a matrix $$D$$ is a diagonal matrix if and only if $$D_{i,j} = 0$$ for all $$i \neq j$$. The notation $$diag(v)$$ is used to denote a diagonal matrix whose main diagonal is the vector $$v$$. Note that not all diagonal matrices need to be square.

Diagonal Matrix

A computational model represents the three words in the sentence 'AI models learn' as three distinct numerical vectors: `v1 = [0.1, 0.5]`, `v2 = [0.9, 0.2]`, and `v3 = [0.4, 0.7]`. To process these words as a single, ordered sequence, how should these vectors be organized into a single matrix?

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