paper
... Consequently, D , DX = DY , and P = D1/2 UΣUT D1/2 . Symmetric channels2 are closely related to conforming probability distributions. We shall illustrate this relation in the next lemma and in Section IV. Lemma 1. If P is conforming, then the corresponding conditional distribution matrix PY |X is po ...
... Consequently, D , DX = DY , and P = D1/2 UΣUT D1/2 . Symmetric channels2 are closely related to conforming probability distributions. We shall illustrate this relation in the next lemma and in Section IV. Lemma 1. If P is conforming, then the corresponding conditional distribution matrix PY |X is po ...
1.2 row reduction and echelon forms
... Any nonzero matrix may be row reduced (that is, transformed by elementary row operations) into more than one matrix in echelon form, using different sequences of row operations. However, the reduced echelon form one obtains from a matrix is unique. The following theorem is proved in Appendix A at th ...
... Any nonzero matrix may be row reduced (that is, transformed by elementary row operations) into more than one matrix in echelon form, using different sequences of row operations. However, the reduced echelon form one obtains from a matrix is unique. The following theorem is proved in Appendix A at th ...
§13. Abstract theory of weights
... With somewhat more labor one can calculate explicitly the λi in terms of the αj . This information is listed in Table 1, for the convenience, although strictly speaking we shall not need it in what follows. The exact structure of the fundamental group can be found by computing elementary divisors, o ...
... With somewhat more labor one can calculate explicitly the λi in terms of the αj . This information is listed in Table 1, for the convenience, although strictly speaking we shall not need it in what follows. The exact structure of the fundamental group can be found by computing elementary divisors, o ...
9. Numerical linear algebra background
... vector-vector operations (x, y ∈ Rn) • inner product xT y: 2n − 1 flops (or 2n if n is large) • sum x + y, scalar multiplication αx: n flops matrix-vector product y = Ax with A ∈ Rm×n • m(2n − 1) flops (or 2mn if n large) • 2N if A is sparse with N nonzero elements • 2p(n + m) if A is given as A = ...
... vector-vector operations (x, y ∈ Rn) • inner product xT y: 2n − 1 flops (or 2n if n is large) • sum x + y, scalar multiplication αx: n flops matrix-vector product y = Ax with A ∈ Rm×n • m(2n − 1) flops (or 2mn if n large) • 2N if A is sparse with N nonzero elements • 2p(n + m) if A is given as A = ...
Fraction-free matrix factors: new forms for LU and QR factors
... This factoring is modeled on other fraction free definitions, such as pseudo-division, and the idea is to inflate the given object or matrix so that subsequent divisions are guaranteed to be exact. However, although this model is satisfactory for pseudo-division, the above matrix factoring has two u ...
... This factoring is modeled on other fraction free definitions, such as pseudo-division, and the idea is to inflate the given object or matrix so that subsequent divisions are guaranteed to be exact. However, although this model is satisfactory for pseudo-division, the above matrix factoring has two u ...
20. Cyclotomic III - Math-UMN
... p would contain a primitive fifth root of unity, so have order divisible by 5. If it had a quadratic factor then × p2 would contain a primitive fifth root of unity, so have order divisible by 5.) Recall the isomorphisms ...
... p would contain a primitive fifth root of unity, so have order divisible by 5. If it had a quadratic factor then × p2 would contain a primitive fifth root of unity, so have order divisible by 5.) Recall the isomorphisms ...
