Linear Algebra Flashcards
(21 cards)
In SVD how can you find the orthogonal basis for the row space of A
The row space is shown by the columns of Q1 that correspond to the non zero eigenvalues of AtA
In SVD how can you find the orthogonal basis for the null space of A
The null space is given by the columns of Q1 that correspond to the zero eigenvalues of AtA
Method for SVD
Just finding the eigenvectors of AtA and AAt and putting sqrt of eigenvalues on leading diagonal.
What is full rank
All the columns/rows in the matrix are linearly independent
What is the rank of a matrix
The number of linearly independent column vectors in the matrix, OR the number of linearly independent row vectors in the matrix.
Relation of a matrices rank to it’s row echelon form
Equal to the number of non zero rows in the row echelon matrix
What does the pseudo inverse do
If matrix A maps r-dimensional row space to r-dimensional column space. A+ is the matrix which inverts this
A+A x = x
A+ b is in the row space for any b
Is A = Q1 Σ Q2t what does A+ equal
A+ = Q2 Σ+ Q1t
What is a rank deficient matrix
One that isn’t full rank
Befit of using pseudo inverse with ill conditioned least squares problem
It enhances the stability of the calculations. For example a small deviation in a value in b could otherwise cause a large change in the solution, but this no longer happens when using pseudo inverse.
How to find l1 norm of a vector
Sum the magnitude of each term
How to find the l2 norm of a vector
Sum the squares of the values in the matrix and sqrt
Find the l infinity norm of a matrix
The maximum individual value in the vector
Definition of a matrix norm
Norm of Ax divided by norm of x (for all x)
What condition number implies problem is well conditioned
Order 20-50
What condition number magnitude implies an ill conditioned problem
Much more than 100
Condition for a matrix A to be positive definite
All the eigenvalues of A are positive
The determinant of all the principal minors are positive
xt A x is positive for all non zero x
Condition for a matrix A to be positive semi-definite
xt A x is positive for all non zero x
What can we do as an alternative to LU decomp when A is a symmetric positive definite matrix
A = LLt
What does a high condition number make the fitting problem sensitive to
Perturbations or round off errors in the right hand side (b)
What’s the effective rank of a matrix
Found the same way as rank, number of non zero eigenvalues values, but you can neglect v small eigenvalues, thus reducing the rank.
