Chapter 2 Topics from Linear algebra

Question 1

What does linearly independent mean

Answer 1

If vectors v1….vn are linearly independent the equation : x1v1+x2v2+….+xnvn = 0 has no solution except that x1=x2=…=xn=0

Question 2

How to show vectors are linearly independent

Answer 2

Write x1v1+x2v2+….+xnvn = 0 and form three lienar equations and propose a martix A such that Ax=0
We need to show there is only one unique solution so reduce A in REF form and if there are all elading variables or the determinant is not 0 ther eis only one unique solution

Question 3

Define the rank of a matrix

Answer 3

Parameter associated with all matrices. It is the largest K for which the matrix has k linearly independent rows and it is also k which the matrix has k linearly independent columns

Question 4

Define Eigenvalue

Answer 4

In a square matrix A if lamda is an eigenvalue then there exists a non zero column vector v with Av=Lamda. There is also a row vector u with: uA=uLamda

Question 5

Define eigenvector

Answer 5

Non zero column vector v with Av=lamdav is called an eigenvector of A correpsonding to lamda

Question 6

Define row eigenvector

Answer 6

Non zero row vector v with uA=ulamda is called an row eigenvector of A corresponding to lamda

Question 7

Define the characteristic polynomial of a matrix A and what does its degree tell you

Answer 7

det(xI -A) : the degree of this the characteristic polynomial of A is the amount of eigenvalues in the matrix and size n of the matrix.

Question 8

What are properties of similar matrices

Answer 8

Similar matrices have the same eigenvalues

Question 9

Explain diagonalisation

Answer 9

If a matrix had n distinct eignvalues, all eigenvalues then have eigenvectors that are linearly independent. Let x1….xn be the eigenvalues and vi be corresponding eigenvectors so Avi=Xivi. If T is a matrix = (v1, v2,…vn) Then A is similar to using T a diagonal matrix D with eigenvalues on the diagonal.

Question 10

What is a diagonal matrix

Answer 10

Has only zero entries apart from on it’s diagonal

Question 11

When can we compute D if A had eigenvalues with multiplicity 2

Answer 11

If there are two free variables and I can pick two eigenvectors that are linearly independent

Question 12

What can you tell about similarity to a diagonal matrix from a matrix’s eigenvalues

Answer 12

n distinct eigenvalues means A is simialr to D
Non distinct eigenvalues means A may or may nto be similar to D
Can only have A = T-1DT for square matrices

Question 13

What does orthogonal matrix mean

Answer 13

It’s inverse is equals to its transpose. Also A x A^T will give me the identity for an orthogonal matrix

Question 14

What is a symmetric matrix

Answer 14

Transpose is equal to the matrix itself - they are the same

Question 15

Let A and nxn symmetric real matrix - what are its properties

Answer 15

All eigenvalues of A are real numbers
There is an orthogonal martix U with U^T A U = D were D is diagonal (similarity to diagonal matrix with an orthogonal matrix)

Question 16

What matrix U is the orthogonal matrix in the diagonalization of a symmetric matrix

Answer 16

U is matrix of normalised eigenvectors from A the symmetric matrix

Question 17

what is another name for dot product

Answer 17

inner product

Question 18

Whats special about symmetric matrices eigenvectors

Answer 18

their inner product must be zero

Question 19

How is the order of eigenvalues in Matrix D - diagonal matrix from diagonalisation decided?

Answer 19

Depends on order of eigenvectors in T

Question 20

What is a positive definite matrix

Answer 20

Symmetric n x n matrix with all positive eigenvalues

Question 21

What is a positive semi definite matrix

Answer 21

Symmetric n x n matrix with all non negative eigenvalues

Question 22

If A is a real matrix m x n what can we say about AA^t and A^t A

Answer 22

They are symmetric and positive semi definite

Question 23

If multiplication of two matrix P and Q exists. what is special about he eigenvalues of PQ and QP

Answer 23

They are the same except for the number of zeros.

Question 24

What are the singular values of A

Answer 24

The positive square roots of the eigenvalues of S=AA^t. Given the symbol S_i - singular value fo A

Question 25

How can similarity be used practically

Answer 25

Easier to study The powers of matrices A ^k = TD^kT^-1

Question 26

State the SVD theorem

Answer 26

If A is n x n then there exists orthogonal matrices U and V such that (U^T)AV=D where D contains the singular values of A in descending order along the diagonal.

Question 27

Steps for constructing the SVD

Answer 27

1. Find S and T such that S = AA^T AND T=A^TA
2. Find their eigenvalues (they are the same so find for only one)
3. Find orthogonal matrix U that diagonalizes S - to do this find eigenvectors of S and place them in matrix in order from the eigenvalues they correspond to largest to smallest
4. Find orthogonal matrix V that diagonalises T - to do this find eigenvectors of S and place them in matrix in order from the eigenvalues they correspond to largest to smallest
5. Find SV of A - square root of AA^t eigenvalues
6. Check U^TAV= D
7. Writing A = U D V^T

Question 28

What is rank approximation

Answer 28

Finding the matrix closest to my matrix with a different rank.

Question 29

Why is rank approximation relevant to search engines

Answer 29

Reducing size of term by document matrix by sending the small singular values to 0. picked a threshold to get rid of. Term by document becomes UD_0V^T