We know, Linear algebra has a very vast variety of applications like animation, data analysis, networking etc. And, we are bound to perform operations like finding an inverse of some huge matrix. Or, for that matter, even finding really big power of a simple 3×3 matrix. It can become cumbersome to do such a feat manually…(But.. Why do I need to know that calculation… I have a computer!!!)
Imagine yourself in a world devoid of a computer or any calculating machine, or may be your computer/ calculator is not powerful enough to handle that kind of computation. But, you HAVE to find a solution.. Because, probably your life depends on it (Just kidding here :)). Anyways, for some reason, you desperately need to pull off that tough calculation!! Now what???
Here comes the characteristic polynomials to our rescue? Oh.. My superhero.. my Messiah.. The Characteristic polynomials!!! ( Give it a big round of applause please.)
Yes.. It lets you do exactly those herculean feat without bursting your grey cells. ( Thank God!!! We are saved.)
Here is how this special saviour looks like.
Δ(t) = det( tI – A)
Where, Δ(t) is the characteristic polynomial, t is an indeterminate, I is a n-square matrix, and A is a n×n square matrix for which we have to do one of those herculean feat.
Friends, this introduction was necessary because we would have to study Caley Hamilton theorem and eigenvectors which would help us in diagonalisation of matrices. And we know, diagonal matrices have some nice properties that are easy to work with (Because we love taking shortcuts and getting things done really fast).
In the next post (Yesss.. In the next post dear friend.. just to keep this post short and simple, before your mind wanders off), we will do some practice problems on finding characteristic polynomial. Watch out this space for more updates. Till then, explore a bit more about it. Have a nice time!!
Folks, how is this post? Comment here to pour your thoughts!