Sum of Squares seminar- Homework 0.
... matrix T = uv > where Ti,j = ui vj .) Equivalently A = U ΣV > where Σ is a diagonal matrix and U and V are orthogonal matrices (satisfying U > U = V > V = I). If A is symmetric then there is such a decomposition with ui = vi for all i (i.e., U = V ). In this case the values σ1 , . . . , σr are known ...
... matrix T = uv > where Ti,j = ui vj .) Equivalently A = U ΣV > where Σ is a diagonal matrix and U and V are orthogonal matrices (satisfying U > U = V > V = I). If A is symmetric then there is such a decomposition with ui = vi for all i (i.e., U = V ). In this case the values σ1 , . . . , σr are known ...
Induction and Mackey Theory
... on W coincide. Denote the elements of W inside V by the formal symbols 1⊗w, w ∈ W . Since V is a G-representation, the group G acts on those and we denote the element g · (1 ⊗ w) ∈ V by the formal symbol g ⊗ w. Since the two H-actions coincide, this must satisfy the rule that h ⊗ w = 1 ⊗ (hw) for al ...
... on W coincide. Denote the elements of W inside V by the formal symbols 1⊗w, w ∈ W . Since V is a G-representation, the group G acts on those and we denote the element g · (1 ⊗ w) ∈ V by the formal symbol g ⊗ w. Since the two H-actions coincide, this must satisfy the rule that h ⊗ w = 1 ⊗ (hw) for al ...
Probabilistically-constrained estimation of random parameters with
... Abstract— The problem of estimating random unknown signal parameters in a noisy linear model is considered. It is assumed that the covariance matrices of the unknown signal parameter and noise vectors are known and that the noise is Gaussian, while the distribution of the random signal parameter vec ...
... Abstract— The problem of estimating random unknown signal parameters in a noisy linear model is considered. It is assumed that the covariance matrices of the unknown signal parameter and noise vectors are known and that the noise is Gaussian, while the distribution of the random signal parameter vec ...
Solutions
... vector subspace of V show that T(V1 ) = {w ∈ R(T) | T(v) = w for some v ∈ V1 } is a vector subspace of W. Solution: We must show that (a) 0W ∈ T(V1 ), (b) w1 + w2 ∈ T(V1 ), for all w1 , w2 ∈ T(V1 ), and (c) cw1 ∈ T(V1 ), for all w1 ∈ T(V1 ) and c ∈ F . (a) Since T is linear T(0V ) = 0W . It must be ...
... vector subspace of V show that T(V1 ) = {w ∈ R(T) | T(v) = w for some v ∈ V1 } is a vector subspace of W. Solution: We must show that (a) 0W ∈ T(V1 ), (b) w1 + w2 ∈ T(V1 ), for all w1 , w2 ∈ T(V1 ), and (c) cw1 ∈ T(V1 ), for all w1 ∈ T(V1 ) and c ∈ F . (a) Since T is linear T(0V ) = 0W . It must be ...
