matlab关于矩阵分解与变换常用命令介绍.ppt

Rank of a Matrix rank(A) (the rank of a m-by-n matrix A) is The maximal number of linearly independent columns =The maximal number of linearly independent rows =The dimension of col(A) =The dimension of row(A) If A is n by m, then rank(A)= min(m,n) If n=rank(A), then A has full row rank If m=rank(A), then A has full column rank Inverse of a matrix Inverse of a square matrix A, denoted by A-1 is the unique matrix s.t. AA-1 =A-1A=I (identity matrix) If A-1 and B-1 exist, then (AB)-1 = B-1A-1, (AT)-1 = (A-1)T For orthonormal matrices For diagonal matrices Dimensions By Thomas Minka. Old and New Matrix Algebra Useful for Statistics Examples / Singular Value Decomposition(SVD) Any matrix A can be decomposed as A=UDVT, where D is a diagonal matrix, with d=rank(A) non-zero elements The fist d rows of U are orthogonal basis for col(A) The fist d rows of V are orthogonal basis for row(A) Applications of the SVD Matrix Pseudoinverse Low-rank matrix approximation Eigen Value Decomposition Any symmetric matrix A can be decomposed as A=UDUT, where D is diagonal, with d=rank(A) non-zero elements The fist d rows of U are orthogonal basis for col(A)=row(A) Re-interpreting Ab First stretch b along the direction of u1 by d1 times Then further stretch it along the direction of u2 by d2 times Low-rank Matrix Inversion In many applications (e.g. linear regression, Gaussian model) we need to calculate the inverse of covariance matrix XTX (each row of n-by-m matrix X is a data sample) If the number of features is huge (e.g. each sample is an image, #sample n#feature m) inverting the m-by-m XTX matrix becomes an problem Complexity of matrix inversion is generally O(n3) Matlab can comfortably solve matrix inversion with m=thousands, but not much more than that Low-rank Matrix Inversion With the help of SVD, we actually do NOT need to explicitly invert XTX Decompose X=UDVT Then XTX = VDUTUDVT = VD2VT Since V(D2)VTV(D2)-1VT=I We know that (XTX )-1= V(D2)-1VT Inver