No One Taught Eigenvalues & EigenVectors Like This

in our high school math we have always been told that if you given a matrix say this a matrix just make a characteristic equation out of it and that will help you find what we call igen values of that Matrix a and then for each igen value just solve this equation to get igen vectors right but we have never been taught what these IG values and igen vectors actually represent in the real world we were simply told to somehow learn learn the steps to calculate them and then forget about it this is the main reason

many people hate maths because they are taught how to calculate things but never why they matter or what they truly mean so are igen values and igen vectors just some fancy numbers and vectors that pop out of an equation or do they have a deeper more intuitive meaning and most importantly can we actually visualize them well in this video we are going to see igen values and igen vectors in action so that they finally make sense assume we have a random 2 cross2 Matrix like this now assume we have a point say 0 comma 1

which we can represent using this Vector now if we multiply this Matrix with this Vector we get another Vector which is this point on the graph and it is represent presented by this Vector so what we saw is this 2 cross2 Matrix ax as something which transforms a vector into some other Vector it is like stretching or shrinking a vector and then rotating it by some amount this means if you take any point on this graph and apply this transformation its new position is calculated by multiplying it with this Matrix so let me do something

interesting here let me take all these points on this graph and applied transformation on all of them at once using this Matrix this is what the transformation will look like the yellow dots are all the new output vectors let us first observe all the points along some line say x AIS you can see that the points along this line are rotated by some angle and also it got a bit stretched right now let us observe all the points along this line or Y AIS where you can see that the points along on this line are

also rotated by some angle and they also got a bit stretched right now here comes the magic this time just observe all of the points along this line the vectors along this line do not rotate at all they only get longer or shorter such lines where the vectors do not rotate but only get Scaled at some point are called igen vectors so these two lines are the igen vectors of this Matrix matx that was mindblowing now for this Matrix one of the igen vectors points diagonally at an angle of 45° meaning it lies along the

line where X is equal to Y this direction stays the same but the vectors along this line get stretched by a factor of three this scaling factor is one of the igen values of this Matrix that means the igen value corresponding to this igen vector is three we also have this other line where X is equal to - y the vectors along this line neither gets rotated nor it gets scaled it Remains the Same meaning the igen value here is one that's what IG values and IG vectors mean in real life but sometimes it can

happen that no real number satisfies the igen value equation what does that mean imagine this Matrix and let us transform all of these points if You observe clearly this Matrix rotates every vector by 90° counterclockwise nothing is stretching or shrinking along any fixed direction if we try to find igen values we end up with an equation that says x * x + 1 is equal to zero but there is no real number that satisfies this equation this means means that there are no real igen values and that tells us something interesting there is no real

direction that remains fixed under this transformation instead of stretching or compressing vectors along a particular line the transformation rotates everything this is a huge difference compared to what we saw earlier let's look at a real world application of them one surprising place where they show up is the page ranking algorithm of Google at least the way it used to work in the 9s assume we have five websites numbered 1 2 3 4 and five now you can see these Rays between different websites like this Ray between 1 and two denotes there is a link on

website one which leads us to website number two similarly you can see array between website 2 and three which denotes there is a link on website two which leads us to website number three and this is how this structure is defined now the most important question here is which website will have more traffic isn't this an interesting problem you might be thinking how igen values and igen vectors might be playing any role here you know that for igen vectors we need a matrix and this is how we will construct this Matrix here we have five

websites which means we will be having a matrix of size five cross five which has five rows and five columns the value at position I and J in The Matrix represents the probability of a user navigating from website I to website J this Matrix is called the Google Matrix or link Matrix and it captures how web traffic flows through links between websites now let's break it down if you look at the First Column there are three out goinging links in website 1 which goes to website 2 3 and four and thus all of them have

a value of 1 over three which means there is a onethird chance of the user navigating from website 1 to website 2 or three or four since we do not have any link from one to itself that is why it has a zero value also there is no outgoing link between website 1 and website 5 and thus this value is also Z similarly look at column two this will be zero because there is no link from two to itself and for the rest of them we have one outgoing link from website 2 to every other

website which means a total of four outgoing links hence we have a value of 1 over4 this pattern continues for all websites now we will compute the igen values and igen vectors of this Matrix we find this nice solution the most important thing to note here is that the largest igen value which is this one gives us the dominant igen Vector which is this this igen Vector tells us the relative importance of each website in the network the website with the highest value in this igen Vector will be the most important website meaning it will

get the most traffic the largest value in this igen Vector is one which is of website 5 and thus it will be ranked one by the Google Now this is the second highest number which is for website 3 and thus website 3 will be ranked second by Google this way we get the following ranking of all the websites this is exactly how Google's page rank algorithm revolutionized web search in its early days this is just one of the many places igen values show up that was simply amazing now now if this video gets 8,000 likes

then I will make another video where I will show more applications of igen vectors and igen values so good [Music]