### A set of pictures that I use for explaining Singular Value Decomposition (SVD)

According to finite-dimensional spectral theorem, a symmetric matrix
has orthonormal eigenvectors, thus Q*Q

^{T}=I and Q^{T}=Q^{-1}. Let us take Q_{1}as eigenvectors of a symmetric matrix formed by A*A^{T}; and Q_{2}as eigenvectors of a symmetric matrix formed by A^{T}*A. According to the (not too involved) proof, SVD decomposes matrix A in a following way: A = Q_{1}*S*Q_{2}^{T}, with eigenvalues being stored in S. As eigenvalues are stored in descending order, the matrix is now effectively multilayered (see pictures above), with a possibility of “peeling” off the more important vectors at first and discarding the less influential information from the latter part of decomposition.
## No comments:

Post a Comment