word
... The scalar product of a real number x with A is denoted xA , and has ij th entry ( xA) ij x Aij . The transpose of A, denoted AT , is the matrix T whose ij th entry is the ji th entry of A: Tij A ji . In other words, transposition “flips” the rows and columns of A. The above operations can be ...
... The scalar product of a real number x with A is denoted xA , and has ij th entry ( xA) ij x Aij . The transpose of A, denoted AT , is the matrix T whose ij th entry is the ji th entry of A: Tij A ji . In other words, transposition “flips” the rows and columns of A. The above operations can be ...
Matrix inversion
... o Matlab usages: s = svd(H) gives the singular values of H. [R, S, Q] = svd(H) gives all 3 matrices of the decomposition. o Applications of SVD: Norm of a matrix: ||A||2 = 1 (Largest singular value) Rank of a matrix: equal to the number of non-zero singular values. Condition number = max ...
... o Matlab usages: s = svd(H) gives the singular values of H. [R, S, Q] = svd(H) gives all 3 matrices of the decomposition. o Applications of SVD: Norm of a matrix: ||A||2 = 1 (Largest singular value) Rank of a matrix: equal to the number of non-zero singular values. Condition number = max ...
section 2.1 and section 2.3
... The determinant of an nn matrix A can be computed by multiplying the entries in any row (or column) by their cofactors and adding the resulting products; that is, for each 1 i, j n det(A) = a1jC1j + a2jC2j +… + anjCnj (cofactor expansion along the jth column) and det(A) = ai1Ci1 + ai2Ci2 +… + a ...
... The determinant of an nn matrix A can be computed by multiplying the entries in any row (or column) by their cofactors and adding the resulting products; that is, for each 1 i, j n det(A) = a1jC1j + a2jC2j +… + anjCnj (cofactor expansion along the jth column) and det(A) = ai1Ci1 + ai2Ci2 +… + a ...
Some algebraic properties of differential operators
... Let K be a differential field with derivation ∂ and let K[∂] be the algebra of differential operators over K. First, we recall the well-known fact that the ring K[∂] is left and right Euclidean, hence it satisfies the left and right Ore conditions. Consequently, we may consider its skewfield of frac ...
... Let K be a differential field with derivation ∂ and let K[∂] be the algebra of differential operators over K. First, we recall the well-known fact that the ring K[∂] is left and right Euclidean, hence it satisfies the left and right Ore conditions. Consequently, we may consider its skewfield of frac ...
Notes on simple Lie algebras and Lie groups
... group G which does not have nontrivial connected (analytic) normal Lie subgroups. Note: Under this definition, the one-dimensional Lie group is not considered to be simple. The above Lemma implies that G is a simple Lie group if and only if g is a simple Lie algebra over R. The following theorem may ...
... group G which does not have nontrivial connected (analytic) normal Lie subgroups. Note: Under this definition, the one-dimensional Lie group is not considered to be simple. The above Lemma implies that G is a simple Lie group if and only if g is a simple Lie algebra over R. The following theorem may ...
Sketching as a Tool for Numerical Linear Algebra
... • Want to show |SAx|2 = (1±ε)|Ax|2 for all x • Can assume columns of A are orthonormal (since we prove this for all x) • Claim: SA is a k x d matrix of i.i.d. N(0,1/k) random variables – First property: for two independent random variables X and Y, with X drawn from N(0,a2 ) and Y drawn from N(0,b2 ...
... • Want to show |SAx|2 = (1±ε)|Ax|2 for all x • Can assume columns of A are orthonormal (since we prove this for all x) • Claim: SA is a k x d matrix of i.i.d. N(0,1/k) random variables – First property: for two independent random variables X and Y, with X drawn from N(0,a2 ) and Y drawn from N(0,b2 ...
The Power of Depth 2 Circuits over Algebras
... identity testing of even width-2 commutative ABP’s. The following result justifies this. C OROLLARY 4. Identity testing of depth 3 circuits (ΣΠΣ) reduces to that of width-2 ABPs. We mentioned before the prospect of using algebra structure results to solve PIT for depth 2 circuits over algebras. Our ...
... identity testing of even width-2 commutative ABP’s. The following result justifies this. C OROLLARY 4. Identity testing of depth 3 circuits (ΣΠΣ) reduces to that of width-2 ABPs. We mentioned before the prospect of using algebra structure results to solve PIT for depth 2 circuits over algebras. Our ...
