The Applications of Matrices | What I wish my teachers told me way earlier

Summary

This video delves into the extensive applications of matrices in various fields, from electronics and image processing to computer graphics and network analysis. It explains how matrices can transform vectors, crucial for understanding systems in linear algebra. The script covers real-world uses such as solving complex systems of equations, analyzing population dynamics, and even aiding in criminal investigations through image enhancement. It also touches on the use of matrices in Google's PageRank algorithm, encryption methods, and their foundational role in machine learning and neural networks, demonstrating the profound impact of matrices in technology and security.

Takeaways

• 📚 Matrices are fundamental in various fields such as circuit analysis, image processing, quantum mechanics, and computer graphics.
• 🧩 Initially, matrices may seem boring due to their basic operations like addition and multiplication, which lack immediate context or excitement.
• 🔍 Matrix multiplication with a vector can transform the vector through scaling and rotation, which is a key concept in linear algebra.
• 🌀 Special matrices like the identity matrix only rotate vectors, while others may only scale them, with eigenvectors being unaffected by rotation.
• 🔑 Matrices are used to solve systems of linear equations, where the coefficients and variables are organized into matrices for easier computation.
• 🦠 The script uses a hypothetical zombie outbreak scenario to illustrate how matrices can model and predict the dynamics of a system over time.
• 🔍 Image processing involves manipulating pixel values in an image, represented as matrices, to apply effects like blurring, sharpening, or edge detection.
• 🌐 The adjacency matrix in graph theory represents connections between nodes and can be used to analyze complex networks like social connections or the web.
• 🔑 The power of an adjacency matrix can reveal paths of different lengths between nodes, and the trace of the matrix cube can indicate the presence of triangles in a network.
• 🤖 Applications of matrices extend to machine learning and neural networks, where they are crucial for the training and operation of algorithms.
• 🔒 Historically, matrices have been used in encryption methods like the hill cipher, demonstrating their role in security.

Q & A

What are some fields where matrices are extremely important for understanding or analyzing different systems?

Matrices are used in fields such as image processing, computer graphics, quantum mechanics, the Google PageRank algorithm, and network analysis.

How do matrices transform vectors?

Matrices can scale and rotate vectors. When a vector is multiplied by a matrix, the resulting vector is a transformation of the original vector according to the matrix's properties.

What is an eigenvector and eigenvalue?

An eigenvector is a vector that, when multiplied by a matrix, only changes scale and not direction. The factor by which the vector is scaled is known as the eigenvalue.

How do matrices help solve systems of equations?

Matrices can be used to represent the coefficients and variables in a system of equations. By applying matrix multiplication, one can find the solution to the system, which corresponds to the input vector that results in a given output vector.

What is a Markov matrix and how is it used?

A Markov matrix is a matrix where each column sums to 1 and contains no negative values, representing probabilities of transitioning from one state to another. It's used to model systems that evolve over time, such as population dynamics.

How can matrices be used to analyze a zombie outbreak scenario?

In a zombie outbreak scenario, matrices can model the dynamics of human and zombie populations over time by representing the rates of infection and cure. By applying matrix multiplication iteratively, one can predict the long-term outcomes of the outbreak.

What is the Google PageRank algorithm and how does it relate to matrices?

The Google PageRank algorithm uses Markov matrices to rank websites. It treats outgoing links as probabilities of transitioning from one site to another, influencing the site's ranking based on the 'importance' of the sites it links to.

How are matrices used in image processing to blur an image?

In image processing, matrices can be used to blur an image by applying a kernel matrix over the image matrix. Each pixel in the resulting image is a weighted average of the pixels under the kernel, which smooths transitions between different colors.

How did law enforcement use matrices to identify an attacker in the Reginald Denny case?

In the Reginald Denny case, law enforcement used image processing techniques involving matrices to enhance the quality of live footage and identify a distinctive tattoo, which helped in securing a conviction.

What is an adjacency matrix and how is it used in network analysis?

An adjacency matrix is a matrix that represents a graph, where each row and column corresponds to a node, and the entries indicate the presence or absence of a connection between nodes. It's used to analyze networks by counting paths of different lengths between nodes.

How do matrices contribute to the field of machine learning and neural networks?

Matrices are fundamental in machine learning and neural networks for coding and manipulating data. They allow for efficient computation of linear transformations, which are essential operations in training and applying machine learning models.

Outlines

00:00

📚 The Importance of Matrices in Various Fields

This paragraph introduces the wide-ranging applications of matrices in fields such as image processing, computer graphics, quantum mechanics, and network analysis. It discusses the initial perception of matrices as a boring math subject and contrasts this with their crucial role in real-world systems. The explanation begins with basic matrix operations and progresses to how matrices can transform vectors, highlighting the concept of eigenvalues and eigenvectors. The paragraph also touches on the use of matrices in solving systems of equations, illustrating the process with an example and mentioning the inverse matrix and its role in finding solutions.

05:00

🧟‍♂️ Matrix Applications in Dynamic Systems: Zombie Outbreak Scenario

This section uses a hypothetical zombie outbreak scenario to demonstrate the application of matrices in analyzing dynamic systems that evolve over time. It describes a situation where humans and zombies are quarantined in a school with specific rates of human-to-zombie and zombie-to-human conversions. The paragraph explains how these dynamics can be represented by a Markov matrix, which is then used to predict the long-term outcome of the population. The concept of eigenvectors and eigenvalues is reintroduced to identify the equilibrium state of the system, showing that over time, the populations will converge to a stable ratio.

10:01

🔍 Matrix Use in Image Processing: The Reginald Denny Case

This paragraph delves into the use of matrices in image processing, using the example of blurring an image. It explains how a kernel matrix is used to average pixel values in an image to achieve a blur effect. The explanation extends to how different kernels can be applied for various effects such as sharpening or edge detection. The paragraph concludes with a real-world application of these techniques in identifying a suspect in the Reginald Denny case by analyzing a video and using image processing to identify a distinguishing mark, which led to a conviction.

15:01

🌐 Exploring Networks and Graph Theory with Matrices

The focus of this paragraph is on the application of matrices in network and graph theory. It starts by illustrating how small networks can be intuitively understood but larger ones require mathematical tools for analysis. The concept of an adjacency matrix is introduced, which represents connections between nodes in a network. The paragraph explains how squaring the adjacency matrix can reveal mutual connections or paths of length two between nodes. It also touches on the idea of using matrix powers to find paths of various lengths and how the trace of the matrix cube can indicate the number of triangles in a network.

20:02

🤖 Matrices in Machine Learning and Neural Networks

This paragraph briefly mentions the role of matrices in machine learning and neural networks, highlighting that these technologies are coded and manipulated using matrix math. It also touches on the use of matrices in older encryption methods like the hill cipher, which incorporates matrix operations for encrypting and decrypting messages. The paragraph concludes with a mention of the importance of matrices in various fields, including security, and acknowledges the sponsorship of Dashlane, a password manager that focuses on internet safety and security.

25:02

📢 Conclusion and Call to Action

The final paragraph serves as a conclusion to the video, summarizing the importance and impact of matrices in various fields. It includes a call to action for viewers to like, subscribe, and follow the creator on social media for updates. It also mentions the use of a bell icon to ensure viewers receive notifications for future content.

Keywords

💡Matrices

Matrices are rectangular arrays of numbers, symbols, or expressions, arranged in rows and columns. In the context of the video, matrices are used to perform operations on vectors, which can represent transformations in various systems. The video explains how matrices can scale and rotate vectors, which is fundamental to understanding more complex systems such as those in computer graphics and quantum mechanics. An example from the script is the transformation of a vector through matrix multiplication, resulting in a new vector that represents a scaled and rotated version of the original.

💡Vector

A vector is a quantity that has both magnitude and direction. In the video, vectors are used to represent points in space, and they are manipulated using matrices to perform linear transformations. The script mentions a vector starting at the origin and ending at (1,1), which is then transformed by a matrix to result in a new vector (1,3), demonstrating the vector's change in both magnitude and direction.

💡Eigenvectors and Eigenvalues

Eigenvectors are vectors that, when multiplied by a particular matrix, only change in scale and not in direction. The scale factor by which the eigenvector is multiplied is known as the eigenvalue. In the video, the concept is introduced to explain how certain vectors are only scaled and not rotated when acted upon by a matrix. The script uses the example of a vector that remains unchanged in direction but doubles in length when multiplied by a matrix, identifying it as an eigenvector with an eigenvalue of 2.

💡Systems of Equations

Systems of equations consist of multiple equations with multiple variables. Matrices are instrumental in solving these systems, as they allow for the representation of the system in a compact form and the use of matrix operations to find solutions. The video script illustrates this by placing coefficients in one matrix, variables in another, and using matrix multiplication to find the output vector, which corresponds to the solution of the system.

💡Linear Transformations

Linear transformations are functions that map vectors to vectors while preserving the operations of addition and scalar multiplication. The video emphasizes that matrices perform linear transformations on vectors, such as scaling and rotating, which are fundamental to various applications like computer graphics and image processing. The script provides examples of matrices that either rotate or scale vectors, maintaining the linearity of the transformation.

💡Markov Matrix

A Markov matrix is a type of matrix used in probability theory to model transitions between different states in a system. The video uses a Markov matrix to represent a scenario of a zombie outbreak, where the matrix's columns sum to 1, and it has no negative values, representing the probabilities of humans turning into zombies or zombies being cured. The script demonstrates how the matrix can predict the long-term outcomes of the population dynamics.

💡Google PageRank Algorithm

The Google PageRank algorithm is a way of measuring the importance of website pages. It works by considering the number and quality of links to a page as votes, with more votes indicating a higher value. The video mentions that the PageRank algorithm uses Markov matrices to rank websites by treating outgoing links as transition probabilities. This is an example of how matrices are used in real-world applications beyond traditional mathematical contexts.

💡Image Processing

Image processing involves the manipulation of images using mathematical techniques. In the video, image processing is discussed in the context of using matrices to perform operations like blurring, sharpening, and edge detection. The script explains how a kernel matrix can be used to blur an image by averaging the pixel values in a region, demonstrating the practical application of matrices in digital image manipulation.

💡Kernel

In image processing, a kernel is a small matrix used to apply a specific effect to an image. The video describes how different kernels can be used for various purposes, such as blurring (box blur), sharpening, or edge detection. The script provides an example of a kernel that averages the pixels within a region to create a blurred effect, showcasing the versatility of kernels in altering image properties.

💡Graph Theory

Graph theory is a branch of mathematics that deals with graphs, which are mathematical structures used to model pairwise relations between objects. In the video, graph theory is used to analyze networks, such as social networks or the internet, through the use of matrices. The script introduces the concept of an adjacency matrix, which represents the connections between nodes in a graph, and how matrix operations can reveal properties of the network, like the presence of triangles or the number of mutual connections between nodes.

Highlights

Matrices are crucial in various fields such as circuit analysis, image processing, quantum mechanics, and the Google Page Rank algorithm.

Matrices can transform vectors through scaling and rotation, which is fundamental in linear algebra.

A matrix can represent a system that evolves over time, such as a zombie outbreak model with humans turning into zombies and vice versa.

Markov matrices, which sum to 1 in each column and have no negative values, are used to represent systems in equilibrium.

Eigenvectors and eigenvalues are used to determine the equilibrium state of a system, such as the long-term outcome of a zombie-human population model.

Matrices are instrumental in solving systems of linear equations, which is a common application in circuit analysis.

The Google Page Rank algorithm uses Markov matrices to rank websites based on the probability of transitioning from one site to another.

Image processing techniques using matrices helped identify a suspect in the Reginald Denny case by analyzing a rose tattoo affiliated with a gang.

Digital images can be represented as matrices, allowing for manipulation such as blurring, sharpening, and edge detection.

Different kernels or matrices can be applied to images for various effects, such as Gaussian blur or edge detection.

Matrices are used in computer graphics for geometric transformations and projecting 3D images onto a 2D plane.

In graph theory, adjacency matrices represent connections between nodes and can reveal information about the network's structure.

The power of an adjacency matrix can show the number of paths of a certain length between nodes in a graph.

Machine learning and neural networks heavily rely on matrix operations for their algorithms.

The Hill cipher is an older encryption method that uses matrix operations for encrypting and decrypting messages.

Dashlane, a password manager,赞助商 was highlighted for its role in keeping users secure online with features like password storage, auto-fill, and security breach notifications.

Transcripts

this video is sponsored by dash lane

circuits and electronics image

processing computer graphics quantum

mechanics the Google page rank algorithm

any kind of network other stuff these

are the kinds of things where matrices

are used and extremely important for

understanding or analyzing different

systems and although I can't discuss

everything in one video this should give

you some insight into the applications

of matrices beyond an introductory

course might include now in the

beginning matrices can be one of the

most boring subjects we learn in math

maybe now for everyone but least that's

how it was for me I mean we're told hey

here's how matrix addition works real

simply just some of the corresponding

entries and you have your answer

then multiply a matrix by a single

number is as simple as multiplying every

entry by that value but when it comes to

matrix multiplication we do this weird

row by column dot product multiplication

which some teachers just give no context

to so yeah this isn't the kind of stuff

that makes you in a major in matrix math

anytime soon I mean you might learn more

in high school but overall a lot of it

just isn't that exciting however I

promise matrices are used way more than

you probably think but the first thing

we need to realize is that matrices do

things two vectors don't take this as a

definition cuz it's obviously not but we

do need to see what happens when we

multiply a matrix by a vector for

example a vector that starts at the

origin and ends at 1 comma 1 can be

written in matrix form as shown X

component on the top and Y component on

the bottom and when you multiply by a

2x2 matrix like this through the

multiplication rules we get a new vector

out in this case of 1 comma 3 so we put

a vector in and the matrix scaled and

rotate it to get a new vector out this

is what I mean by the matrix doing

things to the vector and in this case

different inputs will be rotated and

scaled differently which we'll see in a

sec now some matrices are much simpler

like this one here just rotates put a

vector in aka multiplied by the matrix

and out will come the same vector

rotated 90 degrees counterclockwise

this matrix on the other hand will just

scale any vector that goes in comes out

twice as long but most two-by-two

matrices like this when we were

analyzing aren't as simple different

input vectors I'll just put a few here

as an example gets scaled and rotated

differently however the transformations

are all linear as in any vector on the

same line as one of those inputs will be

mapped to a vector on the same line as

the corresponding outputs these linear

transformations are why we called the

first in-depth class on matrices linear

algebra anyways I'm going to redo those

transformations once more but this time

pay attention to this vector here you'll

notice it's the only one that is just

scaled it doesn't rotate at all and this

would happen to any vector on that same

line because of what we just saw any

vector that is only scaled by a matrix

is called an eigen vector of that matrix

and how much the vector is scaled by or

two in this case since the length

doubled is known as the eigen value I'm

not going to go through how to solve for

these but the vocabulary will come up

later now the last thing to mention is

that the first application of matrices

we typically learn is how they help us

solve systems of equations the

coefficients can go into a matrix the

variables go into another and the

outputs go here notice I'm using the

same matrix as the one from the last

slide by the way using the rules of

matrix multiplication you can see that

this and this are the exact same so

really what this is asking is which

input vector does this matrix map to 1

comma 3 well we already saw the answer

to this 1 comma 3 is this output and the

question of which vector will the matrix

map to this involves us just doing the

same transformations as before but

backwards to find the answer is 1 comma

1 that will be our solution going

backwards is like applying an inverse

matrix and when the desired output is

the one being multiplied then the vector

it came from comes out so x equals 1 and

y equals 1 are the solutions if we plug

those into the

some equations than both are satisfied

which is exactly what we were looking

for if you haven't seen three blue one

Browns essence of linear algebra series

definitely check that out this will make

a lot more sense but here we're focused

on applications which we're going to get

to now now in systems of equations get

more complex all we have to do is expand

our matrix and we can analyze the system

with as many variables as we want the

reason matrices are used in circuits and

electronics for example is because these

can be represented by linear equations

in which all the voltages and currents

are the unknown variables when the

circuits get hectic where we don't want

to solve it by hand we can just have a

computer find an inverse matrix and

we'll have our currents and voltages but

that's still not too exciting so what

about a system that continuously evolves

over time like for example let's say

there's a zombie outbreak at the local

high school

pretty standard situation and the place

is quarantined so no one can go in or

out but the zombie infection is

spreading so we've got humans in the

school and zombies but no one is coming

in or out so the population remains the

same now let's say every hour 20% of

humans will turn into zombies due to

being infected there is a cure for the

disease luckily however it's not always

guaranteed to work so we'll say that

every hour 10% of the zombies will

return back to humans at this moment if

there are 150 zombies and 150 humans the

question is what is going to happen in

the long run now we're going to assume

the changes happen in discrete intervals

at the hour so let's see what happens in

the first hour for the humans of the 150

starting out 80% of them are going to

stay human or not become infected but we

also have to add the 10% of the 150

zombies that become cured and turned

back to human

this leaves us with a hundred and

thirty-five humans after that first hour

it went down just a bit for the new

total of zombies we would take the 20%

of humans that got infected plus the 90%

of the 150 zombies that are not cured

giving us a total of of course 165 since

these two numbers add together muster

300 but we want to know what happens

after a long time so we got to keep

going after another hour we write the

same percentages except now the number

of humans is 135 instead of 150 and for

the zombies we got 165 instead of 150

this now puts 125 for the humans and 175

for the zombies so we seem to keep

losing humans but will this continue

well what we have here is some linear

equations that can be represented as a

matrix of those percentages which don't

change this is multiplied by the inputs

hmz or the current population of humans

and zombies at any time and all of this

equals the populations after that given

our this is called a markov matrix by

the way since its column sum to 1 and it

has no negative values but this is like

what we just saw the matrix we have is

going to do stuff to or scale and rotate

the input vector the first input was 150

comma 150 the initial human and zombie

populations and after 1 hour or

multiplication it gets moved to 135

comma 165 but we have to keep going and

apply another transformation sending it

to 125 comma 175 the populations we

found after 2 hours so as we keep

applying these multiplications the real

question is where does this vector go

well let me put a few vectors on the

graph to show this each representing

populations which add to 300 if we do

the matrix multiplication and look at

the transformations you'll notice this

vector or anything on this line stays

put while everything else moves towards

it that vector is an eigenvector of our

matrix the associated eigenvalue is 1

since it doesn't scale and since that

vector doesn't rotate or scale it is the

equilibrium of the system and therefore

the answer to our question after a long

time the populations will settle to

numbers which lie on this line and add

to 300 which would be 100 for the humans

and 200 for the zombies any other

population values we'll just move a

little closer to these after each hour

if you put those values inside the

equations from before you'll see the

output remains 100 and 200 for the

humans and zombies respectively then if

the percentages were to change the

question just comes down to what is the

eigenvector of the new matrix there may

not be a zombie infestation anytime soon

but this kind of math could be used to

analyze how a virus will spread

throughout a population for example and

one of my favorite applications of this

is the Google page rank algorithm which

involves Markov matrices and ranks

websites by treating outgoing links as

probabilities of transitioning from one

site to another for more on that I have

a dedicated video which I'll link below

now moving on here's a happy story not

at all on April 29 1992 a man named

Reginald Denny was beaten nearly to

death live on national TV and this was

just a completely innocent man who had

done nothing wrong you can see Reginald

laying here probably unconscious after

the attack the attack itself can be seen

here on YouTube but in an attempt to

knock an age-restricted like that video

is I'm only showing the portion right

after now the back story here is that

April 29th 92 was the first day of the

Rodney King riots in Los Angeles

Reginald Denny was a truck driver whose

route for the day involved going through

an area where rioting was taking place

which he was not informed about when he

got there he was stopped by rioters

dragged out of his truck and that's when

the beating took place now identifying

who assaulted Denny was not easy since

the quality of the live footage wasn't

amazing but what help law enforcement

confidently identify one of the

attackers with some advanced math to

understand how this was accomplished we

need to first look at what a digital

images a digital image when you look

real closely is just made up of a bunch

of pixels each of a single color those

colors can then be represented by some

numerical value which means like a

square picture made up of a million

pixels 1000 on each edge could be

represented by a thousand by 1,000

matrix where the entries are the color

values of each pixel

working with black-and-white pictures is

much easier though because the black

pixel come you represented with a zero

and a white can be a 1 let's say well

actually work with grayscale images here

though meaning anything in between zero

and one can exist which will correspond

to a different shade of grey so when it

comes to image processing and

manipulation whether it be blurring an

image detecting edges sharpening an

image and so on it all comes down to

manipulating the pixels in a very

specific way to see an example let's

mathematically blur this image of the

number 1 to do so I'm going to make a 3

by 3 matrix where every entry is 1/9

this is known as a kernel and image

processing by the way then what we're

going to do is lay this kernel over our

image matrix and multiply the individual

entries in each square together then add

the results in this case it's just 1

times 1/9 nine times so the sum of all

those is one yes this kernel is really

just finding the average of the pixels

inside it from there we're going to take

that sum of one and set the center pixel

to that color in the new image it just

so happened not to change it's still 1

or white but that won't always be the

case now you'll notice this grid on the

right where the blurred image will go is

the same size as the one on the left to

get the entire blurred image we're just

going to sweep that red section across

the original one when we slide it over

once all those pixels are still 1 so the

average is also 1 and that's where this

new pixel becomes but after sliding over

again the kernel contains a black pixel

so we find the average of 8 ones and a

zero which is about point eight nine

this corresponds to a very light shade

of gray which we will put in that middle

square after sweeping across the image

mapping each new number to the blurred

image these will be the values I know

this method doesn't really account for

the border but for our purposes we're

just going to keep that white now I'll

actually color and the pixels based on

their values and we get a blurred image

of the number one

actually this is extremely blurred

almost beyond recognition but if we put

an outline along the colored region we

can see the one is still kind of there

the reason this is so blurred is because

we're only working with a hundred pixels

but what the kernel did was kind of took

this sharp edge in the original image

between black and white

and smooth or averaged it so we get this

fading from dark to light in the blurred

image the kernel we use represents a

type of blur known as a box blur and

from Wikipedia if you input a picture

with many more pixels then apply the

blur this is the output you get but

there are several other kernels that'll

all accomplish different things a

Gaussian blur also of course blurs the

image but it assigns more weight to the

middle square so dark pixels stay fairly

dark and vice versa there's a sharpen

kernel and there's edge detection

kernels which search for sharp changes

in color you'll notice that all the

numbers in this kernel sum to zero so if

we put it over a section of an image

where all colors are roughly the same

multiplying by these numbers then adding

the results would just yield zero or a

black pixel which is why that

corresponding area is black the only

sections that aren't are where we find

sharp changes in color aka an edge as

another example of edge detection here's

a poorly taken photograph of someone's

arm and by using edge detection

algorithms researchers were able to

identify a region of some kind of

birthmark or tattoo well this is

actually a zoomed in portion of this

image where those men can be seen

beating Reginald any using image

processing techniques similar to what

we've seen one company was able to

determine that this mark was a rose

tattoo affiliated with a certain gang in

Los Angeles and it was this that helped

him eventually secure a conviction of

one of the perpetrators of the attack

now all this may not have involved much

matrix math like we saw earlier but no I

did simplify some things to avoid going

in too much detail and not only image

processing but computer graphics heavily

use matrices with these what we can do

is take geometric data and incorporate

it into a coordinate system we can then

scale rotate reflect shift images and

more through matrix manipulation but

things do get much more complicated like

when you want to project a 3d image into

a 2d plane we can use matrix map to map

the 3d points and find where they would

appear on the flat screen not going to

go into much more detail in that but

again computer graphics are another very

useful application of matrices but for

those wanting some real tangible results

that come from matrix math let's look at

networks and graph theory graphs can

represent a lot of things people and who

they're friends with connections on a

dating app networks of cities and how

they're connected websites and how they

link to each other and so on with small

networks it can be easy to intuitively

understand what's going on like if I

said here's a group of coworkers and

connections represent mutual friendships

it wouldn't be hard to see like this is

the most popular person and this is the

least popular with only one friend if

you had to find how many mutual friends

these two people have no big deal you

can just count and see that's three but

when the networks get more complex we

need mathematical tools to help us

identify key things this could be like

which website should be ranked the

highest on the web which I mentioned

earlier it could be finding who is more

likely to spread a disease and a college

full of students for which people

involved in the 9/11 terrorist attacks

were most critical to the operation and

should be prioritized by law enforcement

yes they actually did this after 9/11

which I have discussed in a previous

video we need some mathematical

techniques in these cases so we can find

things that our eyes aren't always able

to when the connections get this chaotic

but let's do a matrices can reveal when

it comes to dating apps imagine this app

only has three men signed up that will

label 1 through 3 and 3 women labeled 4

through 6 and there

together as shown these are mutual

connections by the way like both people

swiping right and you'll know for now

there are no same-sex matches we see

this first guys matched with all three

women the second guy with two and the

third with one

now this graph provides a nice visual

for this situation but what we can also

do is make a table with six rows and six

columns for the six people and analyze

this instead we'll say if two people are

matched like person one in person for

our then in the square located and in

column 1 and row four we will put a 1

however since these are mutual

connections we need to also include a 1

in column 4 and Row 1 basically if one

matches with 4 then of course 4 has

matched with 1 so the data has to

reflect that so that means yes the

tables going to be symmetric about the

diagonal then if two people are not

connected like person 1 and 2 we'll put

a 0 in that square in this case column 1

in row 2 but of course we can't forget

column 2 and row 1 since no one matches

with themself the diagonal is going to

be all zeros and then these would be the

rest of the connections so if you want

to know whether person 5 and 2 are

connected by the table just go to column

5 and Row 2 or vice versa and see if

there's a one or a zero there from this

there are obvious things we can see like

for person 1 we can look down their

column or across the row and find they

have three matches in total because of

the three ones but we're going to use

some slightly more advanced math to

analyze this graph so instead of

considering this a table we're going to

call it a matrix but it's taking away

the gridlines but otherwise nothing has

changed when it comes to graphs this is

known as an adjacency matrix another way

to interpret this though is that it

tells us how many paths of length one

exists between any two nodes oh and for

the rest of this video when I say path I

just mean any sequence of edges that

joins a sequence of vertices basically

just a walk those no graph theory may

not like this because a path is usually

more specific but I am being generic

here okay so what's this really mean

we'll look at column 6 and row 1

we know this says that those two people

are matched no big deal

but it also means there is one path of

length one that exists between them

those are saying the same thing if we

put a dot at person 1 and can only

traverse one edge well there's one way

to get to person 6 that's what this one

represents on the other hand there are

zero ways to get from person 1 to person

2 in one edge if you start a person 1

there are paths to person 2 but they all

have a length of 2 which is not what we

were looking for but now what if I want

to see quickly how many mutual matches

two people have well if we look at

person 1 and 2 this isn't tough we see

there are 2 mutual matches however this

question of mutual matches is no

different than asking how many paths of

length 2 exists between person 1 and

person 2 well we just saw that the

answer is 2 as expected since it's the

same question again if I start at 1 and

go to 4 than 2 or 5 than 2 those two

paths mean two mutual connections the

cool thing though is that we can find

how many of these paths of length 2

exist between any two nodes by just

multiplying the adjacency matrix by

itself or squaring it we see for person

1 and 2 there are 2 mutual connections

so that checks out then if you look at

the graph for person 2 and 3 they have

no mutual connections and on the matrix

this checks out as well from column 3

and Row 2 having a 0 however person 1

and 3 are both connected to 6 and no one

else which we can also find in the

matrix and you get the idea but now what

would the diagonal mean well that's how

many paths of length 2 it exists between

a person and themselves

aka how many matches they have think

about it for person 1 if I start there

to get back to 1 in two edges I can go

into 4 than 1 5 than 1 or 6 and 1 3

options for the 3 connections this is

why we see a 3 there in the matrix

person 2 that has two matches and it

goes up

then if we multiply the new matrix by

the original the same as finding the

original cubed we get all the paths of

length 3 from one person to another see

the original matrix to some power tells

us how many paths of that length exist

between any two notes now if we have a

same-sex connection and link one to two

let's say all we have to do is add a 1

to the original matrix in the first

column second row and vice versa

see when implementing this in the

software we just have to make small

tweaks to the adjacency matrix and from

there squaring or cubing it tells us a

lot actually something I found

interesting was what the matrix cube

tells us specifically the diagonal for

one it tells us how many paths of length

three exists between a person and

themselves but looking at the graph a

length three path back to yourself tells

us there's a triangle there I'm not

going to explain this one in depth but

if you sum the numbers along the

diagonal also known as the trace of the

matrix and divide by six that tells you

how many triangles in total there are in

the network I just found that to be a

cool thing the matrix tells us which you

wouldn't think about it first then on

another topic something I haven't even

mentioned yet is machine learning and

neural networks which are coded with and

manipulated by matrix math

mathematically matrices are a huge

aspect of what allows the machine to

quote learn or in terms of security

there's an example of an older kind of

encryption method which is the hill

cipher this cipher incorporates matrix

operations in order to encrypt and

decrypt messages although no it's not a

modern encryption method and as much as

I'd love to keep going into depth on

different subjects this video is already

quite long so hopefully this show just

how powerful and impactful matrices are

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