Chapter 2 Topics from Linear algebra Flashcards
What does linearly independent mean
If vectors v1….vn are linearly independent the equation : x1v1+x2v2+….+xnvn = 0 has no solution except that x1=x2=…=xn=0
How to show vectors are linearly independent
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
Define the rank of a matrix
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
Define Eigenvalue
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
Define eigenvector
Non zero column vector v with Av=lamdav is called an eigenvector of A correpsonding to lamda
Define row eigenvector
Non zero row vector v with uA=ulamda is called an row eigenvector of A corresponding to lamda
Define the characteristic polynomial of a matrix A and what does its degree tell you
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.
What are properties of similar matrices
Similar matrices have the same eigenvalues
Explain diagonalisation
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.
What is a diagonal matrix
Has only zero entries apart from on it’s diagonal
When can we compute D if A had eigenvalues with multiplicity 2
If there are two free variables and I can pick two eigenvectors that are linearly independent
What can you tell about similarity to a diagonal matrix from a matrix’s eigenvalues
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
What does orthogonal matrix mean
It’s inverse is equals to its transpose. Also A x A^T will give me the identity for an orthogonal matrix
What is a symmetric matrix
Transpose is equal to the matrix itself - they are the same
Let A and nxn symmetric real matrix - what are its properties
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)
What matrix U is the orthogonal matrix in the diagonalization of a symmetric matrix
U is matrix of normalised eigenvectors from A the symmetric matrix
what is another name for dot product
inner product
Whats special about symmetric matrices eigenvectors
their inner product must be zero
How is the order of eigenvalues in Matrix D - diagonal matrix from diagonalisation decided?
Depends on order of eigenvectors in T
What is a positive definite matrix
Symmetric n x n matrix with all positive eigenvalues
What is a positive semi definite matrix
Symmetric n x n matrix with all non negative eigenvalues
If A is a real matrix m x n what can we say about AA^t and A^t A
They are symmetric and positive semi definite
If multiplication of two matrix P and Q exists. what is special about he eigenvalues of PQ and QP
They are the same except for the number of zeros.
What are the singular values of A
The positive square roots of the eigenvalues of S=AA^t. Given the symbol S_i - singular value fo A
How can similarity be used practically
Easier to study The powers of matrices A ^k = TD^kT^-1
State the SVD theorem
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.
Steps for constructing the SVD
- Find S and T such that S = AA^T AND T=A^TA
- Find their eigenvalues (they are the same so find for only one)
- 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
- 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
- Find SV of A - square root of AA^t eigenvalues
- Check U^TAV= D
- Writing A = U D V^T
What is rank approximation
Finding the matrix closest to my matrix with a different rank.
Why is rank approximation relevant to search engines
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
