Category Archives: Linear Algebra

Why is a² + b² ≥ 2ab ?

This post can be regarded as a sequel to my previous (and very ancient) post on 1+2+3+…. Though these two posts are not quite logically related, they share the same spirit (I’m asking a dumb question again). How can one … Continue reading

Posted in Calculus, Discrete Mathematics, Geometry, Inequalities, Linear Algebra, Probability | 1 Comment

The Cauchy-Schwarz inequality and the Lagrange identity

The classical Lagrange identity is the following: This can be proven by expanding and separating the terms into the cross-terms part and the non cross-terms part. The Lagrange identity implies the Cauchy-Schwarz inequality in . And when , this can … Continue reading

Posted in Algebra, Group theory, Inequalities, Linear Algebra | Leave a comment

Why a vector field rotates about its curl?

Let be a smooth vector field on . We define its curl to be the vector field It is often claimed in advanced calculus that measures how fast “rotates”. More specifically, suppose , then locally around , the vector field … Continue reading

Posted in Calculus, Linear Algebra | 1 Comment

Why does a mirror reverse left and right, but not top and bottom?

If you stand in front of a mirror and raise your right hand, it appears in the mirror that you raise your left hand. Why does a mirror reverse left and right, but not top and bottom? For some of … Continue reading

Posted in Geometry, Linear Algebra, Miscellaneous | Leave a comment

Three by three skew-symmetric matrix

This is just a short remark about 3 x 3 skew-symmetric-matrix. Theorem.          If is nonzero and skew-symmetric, then two of its singular values are nonzero and equal, while the other one is zero. Proof.       … Continue reading

Posted in Linear Algebra | Tagged | Leave a comment

Matrix multiplication as a convolution

1. The product of two power series/polynomials is The coefficients given by is sometimes called the Cauchy product. This is a convolution. 2. Let be a finite group and be the group algebra with complex coefficients. Let be two elements … Continue reading

Posted in Analysis, Fourier analysis, Linear Algebra | Tagged , | Leave a comment

S-Lemma

This post aims at proving a standard result in quadratic programming called S-Lemma. The use of this result will be obvious in a future post. We begin with a lemma. Lemma 1.           Let be two … Continue reading

Posted in Applied mathematics, Linear Algebra, Optimization | 4 Comments