From the course: Machine Learning Foundations: Linear Algebra
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Calculating eigenvalues and eigenvectors
From the course: Machine Learning Foundations: Linear Algebra
Calculating eigenvalues and eigenvectors
- [Instructor] In the complex world of linear algebra, sometimes we discover simple, straight-away techniques that help us solve equations. One of those equations is A multiplied by v equals lambda multiplied by v. This says if we multiply matrix A with some vector v, it is the same as multiplying vector v by some scalar lambda. In case this equation is true, we call we vector v eigenvector and scalar lambda associated eigenvalue of matrix A. We can provide this equation as A multiplied by v minus lambda multiplied by v equals zero. Meaning, if we subtract lambda v from Av, we get a zero vector. Finally, we can write this equation as A minus lambda I multiplied by v equals zero, where I is an identity matrix. At the end, determinant of A minus lambda I will be equal to zero. Let's see how to calculate eigenvalues and eigenvectors in the following example. We have a two-by-two matrix A that has elements 3, 4, minus 1…
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