LU Factorization

LU Factorization is used to factor matrix A as the product of lower-triangular matrix L and upper-triangular matrix U.

A = LU

In order to do such decomposition, one must have the knowledge of Gaussian Elimination. 

To better understand how this works, given the matrix M below, we'll get it's LU Decomposition.

To get U we will do the following transformation: 

A lot of us (or maybe just me) have at least once imagined what it be like to become spy. With their high-tech gadgets, jaw-dropping acrobatic stunts, and, more importantly, a covert life. We often see in movies that they pass around unreadable messages that seems to make no sense. But we eventually see that every message is just puzzle that is yet to be decoded.

Let's have an activity where we will try to decode this message:

The given encoded message was derived from this equation:

In order to get the original message (M),  we can rewrite the equation to:

And we already have an agreement with the sender that our encoding matrix will be:

We need to first solve for the inverse of C. This can be done via LU Decomposition.

Now, it is just a matter of plugging in our values:

The result is represented by their ASCII values, so we can convert each of them.

And we will have the original message,

"Sherman - Morrison Formula!"

Which is actually the name of the formula we used to encode our message!