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In the Unscented Transform video, we've introduced the concept of a Matrix square root.

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And in this video, we're going to dive into a bit more detail about what the Matrix square root actually

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is.

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So mathematically speaking, the square root matrix where B is equal to the square root of A is a matrix

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that satisfies relationship that A is a good A B squared.

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So just like we did with SCALARS, this is what we want to understand, a square root of a matrixes.

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However, we know that matrices don't operate the same way as normal.

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Skylar's is different rules that apply to Matrix's compared to the scale of variables.

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So in the filtering domain, we usually operate on covariance matrices, which are by definition positive

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semi definite.

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So we are going to limit the definition of the square root matrix of B is equal to the square root day

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to be the matrix that satisfies this relationship here that A is equal to be Thom's B transpose.

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So instead of just having it to be a zero base squared, we're going to specifically define it B, B

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times, B transpose.

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And we're going to do this because of two reasons.

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First is that the Matrix square root of Aziga to be squared can be expanded out to B, B times B.

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So obviously for this equation to work, Bayes going to have to be a square matrix.

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We also know that covariance matrix is a positive semi definite and the definition of a positive semi

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definite is a matrix that can be broken down into this relationship here of B transpose times.

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B Now combining these two conditions together, we get that the square root condition of a positive

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70 definite matrix means that the Matrix A can be broken down into B times B transpose.

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So this is the definition that we're going to use for the square root of the covariance matrix.

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So then how do we actually go about calculating the square root of a so how do we actually calculate

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this B matrix?

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And one way of calculating this matrix is to use Sokolsky decomposition.

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And what this actually says is that didkovsky decomposition of a positive semi-finished definite matrix

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A is the decomposition into the form of A is equal to L times l conjugate transpose.

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So here L is a lower triangle matrix and LWR is the conjugate transpose of this matrix.

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L have a we know that we're going to be operating with real numbers.

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So the covariance matrix is going to be a real rather than a complex matrix.

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So when L is a real positive, definite matrix then a factorization can be written just as a transpose

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instead of a conjugate transpose.

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And if you notice from the previous slide, the definition of the square root matrix is basically in

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this form here.

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So using this form here, L is going to be the square root matrix of a.

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So this is saying for the unsane and transform we can use a Korski decomposition to calculate the square

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root of the covariance matrix.

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And this is just going to be by taking the decomposition of the covariance matrix and then the lower

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triad will matrix that the composition gives you is just going to be the square root matrix.

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So this is how we're going to be calculating the square root matrix.

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Whenever we're doing any filtering, we're going to use a koskie decomposition and there's lots of numerical

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libraries that do this automatically.

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So this is a very common numerical operation.