## Todai Entrance Exam: Math 2017 – Problem 1

(1)         Therefore:     (2)     Then we solve the equation for each , we then arrive:     (3)     (4) As:         Therefore:     So:     (5)     Notice:     Therefore and .     (6)     Rayleigh quotient? [...]

## Todai Entrance Exam: Math 2019 – Problem 1

(1)     (2) Prove that if is unitary then is orgthogonal: As , if is unitary:     Then, (because they are the real part) and (because they are the imaginary part). Therefore:     So the statement follows. Prove that if is orthogonal then is unitary: If then as shown above, it must [...]

## Todai Entrance Exam: Math 2018 – Problem 1

Problem link (1) Use Gaussian Elimination to address (i). All the rest can be easily inferred. (2) Proof by contradiction: Assume and the solution of the simultaneous linear equation does not exist. If then . However, and , so is guaranteed to have linearly dependent columns. Therefore, we can always represent as a linear combination [...]

## Proof that projection matrix P has lambda = 0 and lambda = 1

Assume that is an eigenvector of . Then has atmost two eigenvalues (0 or 1).     Prove that there exists an eigenvector of with eigenvalue equals 1: We have , therefore we can pick any column vectors of and it will be an eigenvector of with eigenvalue equals 1. Prove that there exists an [...]

## 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 [...]