Tivadar Danka Profile picture
Mar 3, 2022 13 tweets 5 min read Read on X
There is more than one way to think about matrix multiplication.

By definition, it is not easy to understand. However, there are multiple ways of looking at it, each one revealing invaluable insights.

Let's take a look at them!

↓ A thread. ↓
First, let's unravel the definition and visualize what happens.

For instance, the element in the 2nd row and 1st column of the product matrix is created from the 2nd row of the left and 1st column of the right matrices by summing their elementwise product.
To move beyond the definition, let's introduce some notations.

A matrix is built from rows and vectors. These can be viewed as individual vectors.

You can think of them as a horizontal stack of column vectors or a vertical stack of row vectors.
Let's start by multiplying a matrix and a row vector.

By writing out the definition, it turns out that the product is just a linear combination of the columns, where the coefficients are determined by the vector we are multiplying with!
Taking this one step further, we can stack another vector.

This way, we can see that the product of an (n x n) and an (n x 2) matrix equals the product of the left matrix and the columns of the right matrix, horizontally stacked.
Applying the same logic, we can finally see that the product matrix is nothing else than the left matrix times the columns of the right matrix, horizontally stacked.

This is an extremely powerful way of thinking about matrix multiplication.
We can also get the product as vertically stacked row vectors by switching our viewpoint a bit.
There is another interpretation of matrix multiplication.

Let's rewind and go back to the beginning, studying the product of a matrix 𝐴 and a column vector 𝑥.

Do the sums in the result look familiar?
These sums are just the dot product of the row vectors of 𝐴, taken with the column vector 𝑥!
In general, the product of 𝐴 and 𝐵 is simply the dot products of row vectors from 𝐴 and column vectors from 𝐵!
To sum up, we have three interpretations: matrix multiplication as

1. vertically stacking row vectors,
2. horizontally stacking column vectors,
3. and as dot products of row vectors with column vectors.

When studying matrices, each of them is immensely useful.
Having a deep understanding of math will make you a better engineer. I want to help you with this, so I am writing a comprehensive book about the subject.

If you are interested in the details and beauties of mathematics, check out the early access!

tivadardanka.com/book
If you have enjoyed this thread, consider giving it a retweet and following me!

I regularly post deep-dive explanations about seemingly complex concepts from mathematics and machine learning.

Mathematics is beautiful, and I want to show this to you.

• • •

Missing some Tweet in this thread? You can try to force a refresh
 

Keep Current with Tivadar Danka

Tivadar Danka Profile picture

Stay in touch and get notified when new unrolls are available from this author!

Read all threads

This Thread may be Removed Anytime!

PDF

Twitter may remove this content at anytime! Save it as PDF for later use!

Try unrolling a thread yourself!

how to unroll video
  1. Follow @ThreadReaderApp to mention us!

  2. From a Twitter thread mention us with a keyword "unroll"
@threadreaderapp unroll

Practice here first or read more on our help page!

More from @TivadarDanka

Sep 11
Logistic regression is one of the simplest models in machine learning, and one of the most revealing.

It shows how to move from geometric intuition to probabilistic reasoning. Mastering it sets the foundation for everything else.

Let’s dissect it step by step! Image
Let’s start with the most basic setup possible: one feature, two classes.

You’re predicting if a student passes or fails based on hours studied.

Your input x is a number, and your output y is either 0 or 1.

Let's build a predictive model! Image
We need a model that outputs values between 0 and 1.

Enter the sigmoid function: σ(ax + b).

If σ(ax + b) > 0.5, we predict pass (1).

Otherwise, fail (0).

It’s a clean way to represent uncertainty with math. Image
Read 15 tweets
Sep 8
Matrix multiplication is not easy to understand.

Even looking at the definition used to make me sweat, let alone trying to comprehend the pattern. Yet, there is a stunningly simple explanation behind it.

Let's pull back the curtain! Image
First, the raw definition.

This is how the product of A and B is given. Not the easiest (or most pleasant) to look at.

We are going to unwrap this. Image
Here is a quick visualization before the technical details.

The element in the i-th row and j-th column of AB is the dot product of A's i-th row and B's j-th column. Image
Read 16 tweets
Sep 7
Behold one of the mightiest tools in mathematics: the camel principle.

I am dead serious. Deep down, this tiny rule is the cog in many methods. Ones that you use every day.

Here is what it is, how it works, and why it is essential: Image
First, the story:

The old Arab passes away, leaving half of his fortune to his eldest son, third to his middle son, and ninth to his smallest.

Upon opening the stable, they realize that the old man had 17 camels. Image
This is a problem, as they cannot split 17 camels into 1/2, 1/3, and 1/9 without cutting some in half.

So, they turn to the wise neighbor for advice. Image
Read 18 tweets
Sep 7
The way you think about the exponential function is wrong.

Don't think so? I'll convince you. Did you realize that multiplying e by itself π times doesn't make sense?

Here is what's really behind the most important function of all time: Image
First things first: terminologies.

The expression aᵇ is read "a raised to the power of b."

(Or a to the b in short.) Image
The number a is called the base, and b is called the exponent.

Let's start with the basics: positive integer exponents. By definition, aⁿ is the repeated multiplication of a by itself n times.

Sounds simple enough. Image
Read 18 tweets
Sep 5
In machine learning, we use the dot product every day.

However, its definition is far from revealing. For instance, what does it have to do with similarity?

There is a beautiful geometric explanation behind: Image
By definition, the dot product (or inner product) of two vectors is defined by the sum of coordinate products. Image
To peek behind the curtain, there are three key properties that we have to understand.

First, the dot product is linear in both variables. This property is called bilinearity. Image
Read 16 tweets
Sep 5
The single biggest argument about statistics: is probability frequentist or Bayesian?

It's neither, and I'll explain why.

Deep-dive explanation incoming: Image
First, let's look at what probability is.

Probability quantitatively measures the likelihood of events, like rolling six with a die. It's a number between zero and one.

This is independent of interpretation; it’s a rule set in stone. Image
In the language of probability theory, the events are formalized by sets within an event space.

The event space is also a set, usually denoted by Ω.) Image
Read 34 tweets

Did Thread Reader help you today?

Support us! We are indie developers!


This site is made by just two indie developers on a laptop doing marketing, support and development! Read more about the story.

Become a Premium Member ($3/month or $30/year) and get exclusive features!

Become Premium

Don't want to be a Premium member but still want to support us?

Make a small donation by buying us coffee ($5) or help with server cost ($10)

Donate via Paypal

Or Donate anonymously using crypto!

Ethereum

0xfe58350B80634f60Fa6Dc149a72b4DFbc17D341E copy

Bitcoin

3ATGMxNzCUFzxpMCHL5sWSt4DVtS8UqXpi copy

Thank you for your support!

Follow Us!

:(