In the proof of Theorem 2.6 (here omitted for brevity), it can be seen that Definition 2.12 is needed to show that the function |‖A__‖| in (2.48) satisfies the triangular inequality introduced in Definition 2.11. An important example of induced matrix norm is the p-norm of a matrix ...
vn] are pairwise conjugate with respect to a quadric Q, then the viare principal axis directions of Qand C is a diagonal matrix. One easily checks that for a conic section given by its equation a rotation by an angle α where tan2α=2c12/(c11−c22) turns the coordinate axes into...
be a group, which can typically be a matrix multiplicative group. Then a family of elements (g(s))s∈R of G is a one-parameter subgroup of G if and only if g(0) is the neutral element e of G and for all s, t: g(s).g(t) = g(s+t). When such a subgroup belongs to ...
Proof Let \rho be an irreducible representation of {\mathcal {H}}. Choose \sigma \in \widehat{{\mathcal {K}}} such that \sigma is a subrepresentation of \rho restricted to {\mathcal {H}}. Then [27, Theorem 8.2] shows that \rho is a subrepresentation of {\textrm{Ind}}_{{\...
Norm[{1, 2}] == 〈{1, 2}, {1, 2}〉 True Any positive-definite symmetric n-by-n matrix A can be used to define an inner product. If A is an identity matrix, the inner product defined by A is the Euclidean inner product. ■ A nonstandard inner product on the coordinate vector...
An n×n matrix D=(dij) is said to be a Euclidean distance matrix (EDM) if there exist points p1,…,pn in some Euclidean space such thatdij=‖pi−pj‖2 for all i,j=1,…,n, where ‖‖ denotes the Euclidean norm. p1,…,pn are called the generating points of D and the dimensi...
Thus, we complete the proof of Proposition 3.1. \square 4 The proof of main Theorem In this section, we will prove Theorem 1.1 by deriving a contradiction. We assume that the mean curvature H is not constant. Firstly, we compute \langle R(e_{i},e_{j}) e_{i},e_{j}\rangle acc...
In particular, the matrix completion problem with fixed basis coefficients was studied by Miao et al. [39], who proposed a rank-corrected procedure to generate an estimator using the nuclear semi-norm and established the corresponding non-asymmetric recovery bounds. Very recently, Javanmard and ...
and the scalar-matrix transpose law (5.464)r⊗Vt=r⊗Vt for all r, r1, r2∈ℝ and V∈ℝcn×m. Proof By means of (5.400) and Theorem 5.56, p. 240, (5.465)Br1+r2⊗V=BVr1+r2=BVr1BVr2=Br1⊗VBr2⊗V=Br1⊗V⊕r2⊗V and (5.466)rgyrr1⊗Vr2⊗V=In. It follows...
As another application of this method (also from =-=[LM00-=-]), here is a quick proof of Theorem 3. It is known [Alo86] that if G is a k-regular -expander graph and A is G's adjacency matrix, then the second eigenvalue of A issksfor somesthat depends on k and ...Linial...