Linear Algebra for Machine Learning and Data Science (13)

Determinants and Eigenvectors

In this final week, you will take a deeper look at determinants. You will learn how determinants can be geometrically interpreted as an area and how to calculate determinants of product and inverse of matrices. We conclude this course with eigenvalues and eigenvectors. Eigenvectors are used in dimensionality reduction in machine learning. You will see how eigenvectors naturally follow from the concept of eigenbases.

Learning Objectives

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• Interpret the determinant of a matrix as an area and calculate the determinant of an inverse of a matrix and a product of matrices.
• Determine the bases and span of vectors.
• Find eigenbases for a special type of linear transformations commonly used in machine learning.
• Calculate the eigenvalues and eigenvectors of a linear transformation (matrix).

Determinants In-depth

Singularity and rank of linear transformations

Just like singular and non-singular matrices, linear transformations can also be singular or non-singular

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Non-singular vs. singular transformations

One trait of the non-singular transformation is that the resulting points after the multiplication cover the entire plane

With singular transformation, the resulting points do not cover the entire plane

Dimensions and ranks of linear transformations

The non-singular matrix transforms into a matrix, hence covering the whole plane, and has a rank of 2

A singular matrix transforms into a single line that doesn’t cover the whole plane and has a rank of 1

A matrix with all zeros transforms into a single point that doesn’t cover the whole plane and has a rank of 0

Determinant as an area

In linear transformation, a determinant is explained as an area (volume)

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Singular vs. non-singular determinants

Since a singular matrix transforms into a line or a point, the determinant is 0

A non-singular matrix can have a negative determinant and the linear transformation happens in a counter-clockwise order

Determinant of a product

The determinant of a product is simply a multiplication of two determinants

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Determinant of a product

The product of a singular and a non-singular matrix (in any order) is

Answer

Singular

That's correct! Since $\text{det}(AB)=\text{det}(A)⋅\text{det}(B)$, if either A or B has determinant 0, it will vanish the product, therefore $\text{det}(AB)=0$, and the resulting matrix is singular.

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Determinants of inverses

Find the determinants

Answer

Matrix 1

0.2

Correct! The determinant of the matrix is:

det(M) = (0.4⋅0.6) - ((-0.2)⋅(-0.2)) = 0.2

 

Matrix 2

0.125

Correct! The determinant of the matrix is:

det(M) = (0.25⋅0.625)-((-0.125)⋅(-0.25) = 0.125

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When the matrix is convertible (non-singular), the determinant of the inverse is 1 over the determinant of the matrix

 

All the information here is based on the Linear Algebra for Machine Learning and Data Science | Coursera from DeepLearning.AI