## Proof of lower-rank matrix factorization

Let and and  , prove that . We can determine  by applying row operations on . First, we will let the second row of  after the first row operation be: . Next, the third row of  after the second row operation would be: . Soon, we will come to () which is a linear combination [...]

## Entropy 2016, bảng A

It took me quite a lot of time to understand the question, though. I do not post the test here as I am not sure if I am permitted to do that. This post serves for my personal use. Even though the original questions are written in Vietnamese, I will be writing my solutions in [...]

## Proof that if all pivots of A are positive then A is a positive definite matrix

Hinted from Math 2270 – Lecture 33 : Positive Definite Matrices, by Dylan Zwick, foot note of page 4. If is symmetric then is always diagonalizable: , . Set (), we have:     Hinted from Introduction to Linear Algebra – Gilbert Strang [WORKING AREA] We have:     Now consider the expression , with is the entry [...]

## Why do the eigenvalues of a triangular matrix lie on its diagonal?

Suppose is a triangular matrix, so is invertible. Now with eigenvalues finding process, we can only shift each ‘s diagonal entry for a to make non-invertible. And the only possible way is to shift each ‘s diagonal entry for a that is also an ‘s diagonal entry, so after that one diagonal entry of will equal [...]

## Proof that unless projection matrix P = I, P is singular

We have: , first we need to prove that : Set , hence each column vector of is a linear combination of column vectors of , so . Here is an by matrix while is an by matrix and column vectors of must be linearly indepedent (so ), hence . If then it’s trivial to show that is [...]

## Proof that non-singular matrix A and singular matrix B add to non-singular matrix C

We have:     A note here is that where is not equivalent to . Consider the two respective column vectors and , for the sake of simplicity and are now scalars, we have but . Because is non-singular and suppose are by matrices, so in first columns are pivots and last columns are free. 10/24/2016: [...]

## [Notes] Reflection through hyperplane

(from Introduction to Linear Algebra [4th Edition] by Gilbert Strang, section 4.4, page 231) I don’t know the official reflection definition in mathematics so it would be a lame to prove that is indeed a reflection matrix, but I will note several stuffs that I have observed. Also special thanks to anyone gave me a [...]

## Proof that if a non-zero vector is orthogonal to a subspace then it is not in the subspace

We consider a non-zero vector and a matrix with set of column vectors is a basis , is orthogonal to . Now we prove that column vectors of are linearly independent. If column vectors of  are linearly independent then is invertible (which means ). We have: . It’s easy to see that  ( is invertible because has linearly [...]