Linear Algebra Flashcards
Define convex
Any line segment joining two points in the curve is above the curve
Singular matrix
Inverse of the matrix doesn’t exist
Multi-variate Gaussian
f(x) = 1/sqrt[ (2 pi)^d |covariance|] exp((x - m).T (x-m)/(2 covariance))
LU Decomposition
QR Decomposition
Singular Value Decomposition
A [m x n] = U [m x m] S [m x n] V.T [n x n]
U, V are orthogonal, unitary
Eigendecomposition
A = Q E Q^-1 for a square matrix A
Columns of Q = eigen vectors
diagonals of E = eigen values
Eigen values and vectors of a symmetric matrix
eigenvals = Real
vectors = orthogonal
Unitary Matrix
Conjugate transpose = inverse
Positive definite vs positive semi-definite matrices
Symmetric Matrix ‘A’ is positive definite if
z.T A z > 0 for every non-zero vector z
Semi-definite: >=0
How to know if a matrix is invertible
Lowest eigen val is positive
SVD and Rank of matrix
Rank of matrix = # of non-zero singular values
When does Ax = b have a unique solution? (hint ranks)
When rank[A] = rank[A| b] = n
where A is m x n, b is m x 1
Dot product vs Cross product
Dot product yields a scalar: A.B = ||A|| ||B|| cos alpha
Cross product yields another vector perpendicular to both A and B with magnitude: A x B = || A || || B || sin alpha
inverse of a 3x3 matrix
1/|A| adjugate (A)
adjugate = transpose of cofactor
cofactor of x_ij = (-1) ^ (i + j) det(of matrix skipping row_i + col_j)
