ppt - IBM Research
... Best Low-Rank Approximation • For any matrix A and integer k, there is a matrix Ak of rank k that is closest to A among all matrices of rank k. • Since rank of Ak is k, it is the product CDT of two k-column matrices C and D – Ak can be found from the SVD (singular value decomposition), where C and ...
... Best Low-Rank Approximation • For any matrix A and integer k, there is a matrix Ak of rank k that is closest to A among all matrices of rank k. • Since rank of Ak is k, it is the product CDT of two k-column matrices C and D – Ak can be found from the SVD (singular value decomposition), where C and ...
DISCRIMINANTS AND RAMIFIED PRIMES 1. Introduction
... Therefore we need to show for any prime number p and prime-power ideal pe such that pe |(p) that discZ/pZ (OK /pe ) is 0 in Z/pZ if and only if e > 1. (Recall that the vanishing of a discriminant is independent of the choice of basis.) Suppose e > 1. Then any x ∈ p − pe is a nonzero nilpotent elemen ...
... Therefore we need to show for any prime number p and prime-power ideal pe such that pe |(p) that discZ/pZ (OK /pe ) is 0 in Z/pZ if and only if e > 1. (Recall that the vanishing of a discriminant is independent of the choice of basis.) Suppose e > 1. Then any x ∈ p − pe is a nonzero nilpotent elemen ...
MTH6140 Linear Algebra II 6 Quadratic forms ∑ ∑
... let Q be the quadratic form represented by A. Then we are told that there are linear functions y1 , . . . , yn and z1 , . . . , zn of the original variables x1 , . . . , xn of Q such that Q = y21 + · · · + y2s − y2s+1 − · · · − y2n = z21 + · · · + z2s0 − z2s0 +1 − · · · − z2n . Now consider the equa ...
... let Q be the quadratic form represented by A. Then we are told that there are linear functions y1 , . . . , yn and z1 , . . . , zn of the original variables x1 , . . . , xn of Q such that Q = y21 + · · · + y2s − y2s+1 − · · · − y2n = z21 + · · · + z2s0 − z2s0 +1 − · · · − z2n . Now consider the equa ...
Markov Chain Monte Carlo, Mixing, and the Spectral Gap
... where σ(T ) is the spectrum of T . Theorem 1 (Perron-Frobenius). Let T be a linear transformation from a finite dimensional vector space V → V . If T is irreducible, aperiodic, and contains all real entries, then R(T ) = σ is an eigenvalue with a positive eigenvector, and if µ 6= σ is an eigenvalue ...
... where σ(T ) is the spectrum of T . Theorem 1 (Perron-Frobenius). Let T be a linear transformation from a finite dimensional vector space V → V . If T is irreducible, aperiodic, and contains all real entries, then R(T ) = σ is an eigenvalue with a positive eigenvector, and if µ 6= σ is an eigenvalue ...
Sections 3.4-3.6
... Note that one check the two closure requirements at once by verifying the following property, called the closure under linear combination: C0. cx + dy V whenever x, y V and c, d . Definition Vector spaces that are important in DEs (as well as other branches of mathematics) are function spaces. ...
... Note that one check the two closure requirements at once by verifying the following property, called the closure under linear combination: C0. cx + dy V whenever x, y V and c, d . Definition Vector spaces that are important in DEs (as well as other branches of mathematics) are function spaces. ...
Vectors and Matrices in Data Mining and Pattern Recognition
... smaller than 10−308 , then a floating point exception called underflow occurs. Similarly, the computation of a floating point number of magnitude larger than 10308 results in overflow. Example 1.4 (floating point computations in computer graphics). The detection of a collision between two three-dimension ...
... smaller than 10−308 , then a floating point exception called underflow occurs. Similarly, the computation of a floating point number of magnitude larger than 10308 results in overflow. Example 1.4 (floating point computations in computer graphics). The detection of a collision between two three-dimension ...