THE DIRECTION THAT DOES NOT CHANGE DIRECTION

ADVT: SMART DOOR LOCK FROM AMAZON

The article has some math. But it will be fun and exciting.

Consider the point P(2,3) in the graph. It has x-coordinate 2 and Y-coordinate 3. We can draw an arrow mark from the origin to the point and call it a vector and represent it by (2,3).
Multiply the vector by this matrix.

The product yields a vector (-3,2). Mark it in the graph. Note, the original vector is rotated by 90 degrees. A matrix can rotate, shear, scale, unshape a vector, or a point in the graph. Hence this technique is widely used in computer graphics.
Now consider the vector (0,0) or origin and multiply it by the same matrix.
[0 -1 * [0 = [0
1 0] 0] 0]

The vector is not distributed. That is, the origin cannot be rotated. It is common sense but an important point. We call this vector (0,0) as "eigenvector' of matrix [0 -1],[1 0].

"The eigenvector is one the direction of which cannot be changed by that matrix".The eigenvector of the matrix
[-6 3 is [1
4 5] 4]
If you matrix-multiply both, you get(6, 24). If you simply multiply (1,4) by 6, you get again (6, 24). It confirms that the vector is simply 'scaled six times' but its direction remains the same. You can mark them in a graph and find it is true.

The value 6 is called 'eigenvalue'. The eigenvalue is the scaling factor.

FINDING EIGEN VECTOR:
Also verifying the above case:

Let (x,y) be an eigenvector of our matrix (-6,3
4,5). 6 is the eigenvalue.
The principle:
Matrix *eigen vector = Eigen value * Eigen vector

Hence, (-6 3 *(x = 6*(x
4 5) y) y)
Matrix-multiplying, we get 2 equations:
-6x+3y =6x
4x +5y =6y
Rearranging:
-12x +3y =0
4x-1y =0
Solving we get, x =1 and y =4
or the eigenvector is (1,4), We got it! The eigenvector principle vector is widely used in google search algorithms, computer graphics, statistics, etc.

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