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Tag: Linear Algebra

Kronecker sum
I’m working on a project these days where I’ve used four different kinds of matrix product, which made me wonder if there’s another kind of product out there that I could find some use for. In the process of looking around for other matrix products, I ran across the Kronecker …
How to Set Axis for Rows and Columns in NumPy
NumPy arrays provide a fast and efficient way to store and manipulate data in Python. They are particularly useful for representing data as vectors and matrices in machine learning. Data in NumPy arrays can be accessed directly via column and row indexes, and this is reasonably straightforward. Nevertheless, sometimes we …
Convex function of diagonals and eigenvalues
Sam Walters posted an elegant theorem on his Twitter account this morning. The theorem follows the pattern of an equality for linear functions generalizing to an inequality for convex functions. We’ll give a little background, state the theorem, and show an example application. Let A be a real symmetric n×n …
Change of basis and Stirling numbers
Polynomials form a vector space—the sum of two polynomials is a polynomial etc.—and the most natural basis for this vector space is powers of x: 1, x, x², x³, … But the power basis is not the only possible basis, and often not the most useful basis in application. Falling …
Book Review: Linear Algebra and Learning from Data by Gilbert Strang
I’ve been a big fan of MIT mathematics professor Dr. Gilbert Strang for many years. A few years ago I reviewed the latest 5th edition of his venerable text on linear algebra. Then last year I learned how he morphed his delightful mathematics book into a brand new title (2019) …
What Is Argmax in Machine Learning?
Argmax is a mathematical function that you may encounter in applied machine learning. For example, you may see “argmax” or “arg max” used in a research paper used to describe an algorithm. You may also be instructed to use the argmax function in your algorithm implementation. This may be the …
Hadamard product
The first time you see matrices, if someone asked you how you multiply two matrices together, your first idea might be to multiply every element of the first matrix by the element in the same position of the corresponding matrix, analogous to the way you add matrices. But that’s not …
How fast can you multiply matrices?
Suppose you want to multiply two 2 × 2 matrices together. How many multiplication operations does it take? Apparently 8, and yet in 1969 Volker Strassen discovered that he could do it with 7 multiplications. Upper and lower bounds The obvious way to multiply two n × n matrices takes …
Distribution of eigenvalues for symmetric Gaussian matrix
Symmetric Gaussian matrices The previous post looked at the distribution of eigenvalues for very general random matrices. In this post we will look at the eigenvalues of matrices with more structure. Fill an n by n matrix A with values drawn from a standard normal distribution and let M be …
Circular law for random matrices
Suppose you create a large matrix M by filling its components with random values. If M has size n by n, then we require the probability distribution for each entry to have mean 0 and variance 1/n. Then the Girko-Ginibri circular law says that the eigenvalues of M are approximately …