QR Decomposition

$$A=QR$$

• This is what makes up QR Decomposition!
• Q is a Orthogonal Matrix. (It has orthonormal column vectors!)
• R is a Upper Triangular Matrix

To get a QR decomposition:

1. Take A, and apply the Gram-Schmidt process to the vectors inside it.
2. Once you have found the orthonormal vectors of A, load them into a matrix by column. You now have Q
3. Now, to find R. This is where some theory comes in.

• We have A and we have Q. We will multiply both sides of this equation to get

$$Q^{\perp }A=Q^{\perp }QR$$

• Now, due to the properties of the orthogonal basis, and the orthonormal columns, we find this

$$Q\ast Q^{\perp }=I$$

• This leaves us with I. The I (identity matrix) is left on the right side!

$$Q^{\perp }A=IR$$

• So, to find R, we just have to solve

$$Q^{\perp }A=R$$

Done! The R matrix has been solved for.

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