# <center><i class="fa fa-edit"></i> Eigen Vector & Eigen Values in Linear Algebra </center>
###### tags: `notes`
:::info
**Goal:**
To understand intuitive term about eigen vector & eigen values
**Resources:**
[Math is Fun](https://www.mathsisfun.com/algebra/eigenvalue.html)
:::

## The Math of it
Linear Algebra is all about matrices and the things we can do with them. One of the operations that we can perform on matrices is multiplying them, it wors a bit differently than what we expect.

For a square matrix <b>A</b>, an Eigenvector and Eigenvalue make this equation true:


We will see how to find them (if they can be found) soon, but first let us see one in action:


Let's do some [matrix multiplies](https://www.mathsisfun.com/algebra/matrix-multiplying.html) to see what we get.


Let's see how we multiply matrix. We can either :
- Multiply a matrix with a number, this is called " Scalar Multiplication"
- Multiply a matrix with another one, this is called a "dot product"

### Multiplying a matrix with a number
Look at the image below, we're multiplying the matrix with a number, all the values are simply multiplied 2 or "scaled" by 2. (Which is why we call this scalar multiplication)


This is the final result :point_down:


### Multiplying a matrix with a matrix
Here we take the first row of the matrix and second column of the second matrix, multiply the corresponding values and add them.


By doing this, we will get the top left value of the resultant matrix. Now we do the same except this time we take the second column of the second matrix.


Now the first row of the resultant matrix is complete. We will now take the second row and first column of the first and second matrix respectively and repeat the process.


Do the same for the last value.


We've successfully multiplied 2 matrices.


## How do we find these eigen things?
We start by finding the <b>eigenvalue</b>: we know this equation must be true:


Now let us put in an [identity matrix](https://www.mathsisfun.com/algebra/matrix-types.html) so we are dealing the matrix vs matrix:


Bring all to left hand side:


if <b>v</b> is non-zero then we can solve for λ using just the [determinant](https://www.mathsisfun.com/algebra/matrix-determinant.html):


Here the example from our previous example:


Now we know <b>eigenvalues</b>, let us find their matching <b>eigenvectors.</b>