In the Unscented Transform video, we've introduced the concept of a Matrix square root.
And in this video, we're going to dive into a bit more detail about what the Matrix square root actually
is.
So mathematically speaking, the square root matrix where B is equal to the square root of A is a matrix
that satisfies relationship that A is a good A B squared.
So just like we did with SCALARS, this is what we want to understand, a square root of a matrixes.
However, we know that matrices don't operate the same way as normal.
Skylar's is different rules that apply to Matrix's compared to the scale of variables.
So in the filtering domain, we usually operate on covariance matrices, which are by definition positive
semi definite.
So we are going to limit the definition of the square root matrix of B is equal to the square root day
to be the matrix that satisfies this relationship here that A is equal to be Thom's B transpose.
So instead of just having it to be a zero base squared, we're going to specifically define it B, B
times, B transpose.
And we're going to do this because of two reasons.
First is that the Matrix square root of Aziga to be squared can be expanded out to B, B times B.
So obviously for this equation to work, Bayes going to have to be a square matrix.
We also know that covariance matrix is a positive semi definite and the definition of a positive semi
definite is a matrix that can be broken down into this relationship here of B transpose times.
B Now combining these two conditions together, we get that the square root condition of a positive
70 definite matrix means that the Matrix A can be broken down into B times B transpose.
So this is the definition that we're going to use for the square root of the covariance matrix.
So then how do we actually go about calculating the square root of a so how do we actually calculate
this B matrix?
And one way of calculating this matrix is to use Sokolsky decomposition.
And what this actually says is that didkovsky decomposition of a positive semi-finished definite matrix
A is the decomposition into the form of A is equal to L times l conjugate transpose.
So here L is a lower triangle matrix and LWR is the conjugate transpose of this matrix.
L have a we know that we're going to be operating with real numbers.
So the covariance matrix is going to be a real rather than a complex matrix.
So when L is a real positive, definite matrix then a factorization can be written just as a transpose
instead of a conjugate transpose.
And if you notice from the previous slide, the definition of the square root matrix is basically in
this form here.
So using this form here, L is going to be the square root matrix of a.
So this is saying for the unsane and transform we can use a Korski decomposition to calculate the square
root of the covariance matrix.
And this is just going to be by taking the decomposition of the covariance matrix and then the lower
triad will matrix that the composition gives you is just going to be the square root matrix.
So this is how we're going to be calculating the square root matrix.
Whenever we're doing any filtering, we're going to use a koskie decomposition and there's lots of numerical
libraries that do this automatically.
So this is a very common numerical operation.