## Introduction to Singular Value Decomposition using Python/Numpy examples and drawings

For that reason, here are several examples showing how the space can be transformed by 2D square matrices. We can see that the two base vectors are not necessarily rotated the same way. Instead of doing the transformation in one movement, we decompose it in three movements. Parameters ---------- matrix : array-like 2D matrix to apply to the unit circle. We will take a rescaled circle the one from the example 3.

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## Introduction to Singular Value Decomposition using Python/Numpy examples and drawings

Note that we took only square matrices. Since the beginning of this series, I emphasized the fact that you can see matrices as linear transformation in space. When you apply this matrix to a vector or to another matrix you will apply this linear transformation to it. We can convert our angle from degrees to radians with the function np. Note that these examples used diagonal matrices all zeros except the diagonal. This transformation can be decomposed in three sub-transformations: 1. The general rule is that the transformation associated with diagonal matrices imply only a rescaling of each coordinate without rotation.

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## Introduction to Singular Value Decomposition using Python/Numpy examples and drawings

If you scroll down the page you will see each step. And we saw also that it corresponds to a rescaling without rotation. Visualize the effect of the matrix in 2D. This is related to the sign of the determinent of the matrix see. Every matrix can be seen as a linear transformation You can see a matrix as a specific linear transformation. We can also transform a circle. Summary I hope that you got how vectors and matrices can be transformed by rotating or scaling matrices.

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## Introduction to Singular Value Decomposition using Python/Numpy examples and drawings

We have chosen the length of the vectors from the rescaling weight from example 3 factor 3 and 2 to match the circle. And receive your free +100 pages ebook pdf version of the linear algebra blog articles. It will be useful to visualize the transformations. Figure The figure containing modified unit circle and basis vectors. .

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## Introduction to Singular Value Decomposition using Python/Numpy examples and drawings

The unit circle represents the coordinates of every unit vectors vector of length 1, see. You can see these new matrices as sub-transformations of the space. The unit circle It is then possible to apply a matrix to all these unit vectors to see the kind of deformation it will produce. Note also that a lot of values are needed if we want the connection between the two demi-spheres. Parameters ---------- vecs : array-like Coordinates of the vectors to plot. For instance, a 3 by 2 matrix will map a 2D space to a 3D space. .

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