Matrix Multiplication, Neural Networks, and LLMs

The Big Picture

Matrix multiplication is one of the main computational building blocks in modern machine learning. Neural networks use it to combine inputs with learned weights. Large language models, or LLMs, are stacked neural networks that perform huge numbers of these matrix multiplications to transform text into predictions.

If you understand matrix multiplication as "mixing numbers according to learned recipes," you already have the foundation for understanding why neural networks and LLMs are so compute-heavy.

Three Core Ideas

Matrix Multiplication

A matrix is a grid of numbers. Multiplying matrices is a structured way of combining rows and columns to produce new values.

Neural Network Layers

A layer takes an input vector, multiplies it by a weight matrix, adds a bias, and then applies a non-linear activation.

LLM Computation

LLMs repeat this idea across many layers and many tokens. Attention and feed-forward blocks are both driven by matrix multiplications.

Guided Tour

This page now follows a visual narrative: start with a tiny matrix, see it become a neural layer, then watch the same pattern scale into a transformer answering a question.

1. Inputs

Numbers or token vectors enter the system.

2. Weights

Matrices encode learned recipes.

3. Mixing

Matrix multiplication creates new features.

4. Attention

Tokens compare with other tokens.

5. Response

The model predicts one token at a time.

Plain-English Vocabulary

Token: a small chunk of text the model reads, such as a word, part of a word, punctuation mark, or symbol.

Vector: an ordered list of numbers. In machine learning, a vector is a compact numerical description of something.

Embedding: a word turned into a list of numbers so the model can work with it mathematically.

Projection: multiplying a vector by a matrix to create a new version of that vector.

Comparison: checking how strongly one token should relate to another token.

Attention: the process that lets a token look at other tokens and decide which ones matter most right now.

Feed-forward layer: another learned matrix-based transformation that updates each token after attention.

Prediction: the model's guess for the next token to output.

How They Fit Together

1

Tokenization

What: Your sentence is split into small text pieces called tokens.

How: A tokenizer uses a fixed vocabulary and matches chunks like words, word-parts, punctuation, or symbols, then converts them into token IDs.

2

Embedding Lookup

What: Each token ID becomes an embedding, which is a vector of numbers.

How: The model stores a large embedding table. Looking up a token means selecting that token's learned row, which becomes the starting vector for the token.

3

Projection

What: The model creates new versions of each token vector.

How: It multiplies the token vector by learned weight matrices. Each matrix creates a projected view useful for a different role inside attention.

4

Comparison And Attention

What: The model decides which tokens should influence each other most.

How: It computes similarity scores between projected token vectors, normalizes those scores into weights, and uses the weights to gather more information from the most relevant tokens.

5

Feed-Forward Layer

What: Each token is remixed again after attention.

How: The model applies another learned matrix transformation plus a non-linear activation, allowing it to build richer patterns than matrix multiplication alone.

6

Prediction

What: The model chooses the next token to generate.

How: The final token state is multiplied by another weight matrix to produce scores over the vocabulary. The model selects a likely next token, appends it, and repeats the full process.

Visual 1: Matrix Multiplication As A Mixing Machine

Read left to right. The input numbers flow into a weight matrix. Each output is created by mixing the inputs with a different recipe.

Input Vector

spice = 2

sweet = 3

x

Weight Matrix

1

4

5

2

=

Output Vector

17

14

Each output is a different weighted blend of the same original inputs.

Real-Life Analogy: A Kitchen Recipe Mixer

Imagine a chef tasting two input ingredients, then using recipes to create several sauces. The ingredients stay the same, but each sauce combines them differently.

Input 1
Spice

Input 2
Sweetness

Chef recipes
weights

2 scoops

3 scoops

Sauce A, Sauce B

The chef does not memorize full meals. The chef learns how strongly each ingredient should influence each output.

That is what a weight matrix does inside a neural network: it learns how much each input feature should matter for each output feature.

How Matrix Multiplication Works

Suppose you have a vector of inputs:

[x1, x2]

And a matrix of weights:

[[w11, w12], [w21, w22]]

Multiplying them creates a new vector where each output is a weighted combination of the inputs.

That is the key machine-learning idea: each output is built by mixing input features using learned numbers.

Worked Example

Input vector:

[2, 3]

Weight matrix:

[[1, 4], [5, 2]]

Output:

[2*1 + 3*5, 2*4 + 3*2] = [17, 14]

The network has turned two inputs into two new learned combinations. A real model does this with far larger vectors and matrices.

From Matrices To Neural Networks

1

Input

Turn raw data into numbers, like pixel values or token embeddings.

2

Linear Mix

Multiply by a weight matrix so the model can combine features.

3

Nonlinearity

Apply an activation function so the network can learn more than simple straight-line relationships.

4

Repeat Across Layers

Each layer builds richer features from the one before it.

Why Neural Networks Need It

A neural network learns by adjusting its weight matrices during training. Those matrices are the learned memory of the model.

When you hear that a model has millions or billions of parameters, many of those parameters live inside these matrices.

Inference is then mostly repeated matrix multiplication using the learned weights.

Visual 2: What A Transformer Layer Does

One token vector enters a transformer block. A vector is just a list of numbers describing that token. Matrix multiplication first creates new projected versions of that token, then attention uses those versions to compare tokens, and finally the feed-forward layer remixes the result again.

Input Embedding

A token like "Paris" becomes a vector of numbers. That vector is the embedding.

Projected Views

Matrix multiplication creates new versions of the token so the model can compare roles like asking, matching, and carrying information.

New Token State

The token is updated using attention first, then a feed-forward layer that does another learned matrix remix.

Visual 3: Tokens Looking At Each Other

Attention lets each token decide which other tokens deserve focus. The comparisons behind attention are built from projected embeddings and matrix multiplication.

User asks a question

Why is the sky blue?

Each token becomes a vector

The model turns words into embeddings so math can happen on them.

Attention scores are computed

Matrix multiplication helps compare tokens and decide which ones are relevant to each other.

How LLMs Use The Same Idea

LLMs first break text into tokens. Each token is then converted into an embedding, which is just a vector of numbers. You can think of a vector as the model's numeric handle for that token. From there, the model repeatedly applies matrix multiplications inside transformer blocks.

Two major places this happens:

Attention: the model creates projected versions of token embeddings, then compares tokens to decide what information matters.

Feed-forward layers: after attention, more matrix multiplications transform each token representation again so it carries richer meaning into the next layer.

So even though LLMs feel linguistic, their internal computation is mostly large-scale linear algebra.

Why LLMs Need So Much Hardware

Each token passes through many layers, and each layer uses large matrices. That means a single response can require enormous numbers of multiply-and-add operations.

This is why GPUs are so important: they are good at doing many matrix operations in parallel.

Mini Animated Walkthrough: How An LLM Generates A Response

Use this as a mental movie. Think of the model as repeatedly turning text into embeddings, projecting those embeddings with matrices, comparing tokens through attention, remixing them with a feed-forward layer, and then predicting the next token.

Scene A: Tokenize the question

Why do planes fly ?

Scene B: Turn tokens into vectors

0.8

-0.1

0.4

1.2

0.3

0.9

-0.5

0.2

Scene C: Attention highlights context

The model computes which earlier tokens matter most for the current one.

Scene D: Predict the next word

because their wings ...

What stays constant

The core operation is still matrix multiplication. The model just performs it many times at many sizes.

What changes

The vectors get richer after every layer, because attention and feed-forward blocks keep remixing the information.

Why this helps

Each next-token prediction gets to use context built by many rounds of matrix-based transformation.

Step 1

Your question becomes tokens and vectors

For a prompt like "Why do airplanes fly?", the model splits the text into tokens. Each token then becomes an embedding, which is a vector: a list of numbers representing that token in a learned way.

Step 2

Vectors are projected

Matrix multiplication turns those embeddings into new versions. A projection just means "a matrix-made view of the same vector" that helps with the next job.

Step 3

Comparisons create attention

The model compares projected versions of tokens. The token for "fly" may look toward "airplanes" more strongly than toward "do". Those comparisons create attention, which tells the model what context matters most.

Step 4

Feed-forward layers remix the meaning

After attention, the feed-forward layer applies another learned matrix transformation to each token. This helps the model build a richer internal meaning before it decides what comes next.

Step 5

The model predicts the next token

At the end of the stack, the model scores many possible next tokens. Prediction means choosing the most suitable next token, adding it to the sequence, and then repeating the entire cycle again.

Common Misunderstanding

People sometimes imagine LLMs as mostly "stored text." A better intuition is that they are giant systems of learned weights. The knowledge is not stored like a dictionary entry. It is distributed across many numerical relationships inside weight matrices.

Study Path

Session 1

Practice multiplying a small vector by a small matrix.

Session 2

Learn why a linear layer is written as y = Wx + b.

Session 3

Study activations like ReLU so you see why pure matrix multiplication is not enough.

Session 4

Read a simple introduction to transformers and identify where the matrix multiplications happen.

Quick Self-Test

Why is matrix multiplication useful in neural networks?

Because it lets the model combine many input features into new learned combinations using weights.

What is a layer doing in simple terms?

It takes an input vector, multiplies it by learned weights, adds a bias, and applies an activation.

Where do LLMs use matrix multiplication?

In embedding projections, attention projections, and feed-forward layers throughout the transformer.

Why are GPUs good for LLMs?

Because GPUs are built to run many parallel multiply-and-add operations efficiently.

Good Next Resources

3Blue1Brown linear algebra is excellent for visual intuition around vectors, matrices, and transformations.

3Blue1Brown neural networks helps connect the math to network layers and learning.

The Illustrated Transformer is a strong visual bridge from neural networks to transformers and LLM-style architectures.

Matrix Multiplication to Neural Networks to LLMs