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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 ...
ALTERNATING PROJECTIONS ON NON
... which the usual complex algebraic varieties are special cases). However, algebraic varieties are manifolds except at the singular set (singular locus). Moreover, the singular set is a variety of smaller dimension, and hence makes up a very small part of the original variety. Since Theorem 1.6 is a l ...
... which the usual complex algebraic varieties are special cases). However, algebraic varieties are manifolds except at the singular set (singular locus). Moreover, the singular set is a variety of smaller dimension, and hence makes up a very small part of the original variety. Since Theorem 1.6 is a l ...
AN ASYMPTOTIC FORMULA FOR THE NUMBER OF NON
... We note that δ-smooth margins are also δ 0 -smooth for any 0 < δ 0 < δ. Condition (1.1.3) requires that the entries of the typical matrix are of the same order and it plays a crucial role in our proofs. Often, one can show that margins are smooth by predicting what the solution to the optimization p ...
... We note that δ-smooth margins are also δ 0 -smooth for any 0 < δ 0 < δ. Condition (1.1.3) requires that the entries of the typical matrix are of the same order and it plays a crucial role in our proofs. Often, one can show that margins are smooth by predicting what the solution to the optimization p ...
Section III.15. Factor-Group Computations and Simple
... Z2 and the square of each element (coset) is the identity (H). So H · H = H and (σH) · (σH) = σ2 H = H. So if α ∈ H then α2 ∈ H and if β ∈ / H (then β ∈ σH) then β 2 ∈ H. So, the square of every element of A4 is in H. But in A4 we have (1, 2, 3) = (1, 3, 2)2 and (1, 3, 2) = (1, 2, 3)2 (1, 2, 4) = (1 ...
... Z2 and the square of each element (coset) is the identity (H). So H · H = H and (σH) · (σH) = σ2 H = H. So if α ∈ H then α2 ∈ H and if β ∈ / H (then β ∈ σH) then β 2 ∈ H. So, the square of every element of A4 is in H. But in A4 we have (1, 2, 3) = (1, 3, 2)2 and (1, 3, 2) = (1, 2, 3)2 (1, 2, 4) = (1 ...
On the multiplicity of zeroes of polyno
... an (isolated) root for P (q) if, in the factorization (2.2), there are exactly k quaternions αij which lie on the sphere Sα . Note that in the case of isolated zeroes, this definition does not imply that one can factor (q − α)∗j , and therefore this definition is essentially different from the one s ...
... an (isolated) root for P (q) if, in the factorization (2.2), there are exactly k quaternions αij which lie on the sphere Sα . Note that in the case of isolated zeroes, this definition does not imply that one can factor (q − α)∗j , and therefore this definition is essentially different from the one s ...
Vector Spaces
... Definition 14. A linear combination of vectors v 1 , v 2 , . . . , v n is a vector of the form λ1 v 1 + λ2 v 2 + · · · + λn v n , where λ1 , λ2 , . . . , λn are scalars. Definition 15. The span of the vectors v 1 , v 2 , . . . , v n is the set of all linear combinations of v 1 , v 2 , . . . , v n : ...
... Definition 14. A linear combination of vectors v 1 , v 2 , . . . , v n is a vector of the form λ1 v 1 + λ2 v 2 + · · · + λn v n , where λ1 , λ2 , . . . , λn are scalars. Definition 15. The span of the vectors v 1 , v 2 , . . . , v n is the set of all linear combinations of v 1 , v 2 , . . . , v n : ...
Jordan normal form
In linear algebra, a Jordan normal form (often called Jordan canonical form)of a linear operator on a finite-dimensional vector space is an upper triangular matrix of a particular form called a Jordan matrix, representing the operator with respect to some basis. Such matrix has each non-zero off-diagonal entry equal to 1, immediately above the main diagonal (on the superdiagonal), and with identical diagonal entries to the left and below them. If the vector space is over a field K, then a basis with respect to which the matrix has the required form exists if and only if all eigenvalues of the matrix lie in K, or equivalently if the characteristic polynomial of the operator splits into linear factors over K. This condition is always satisfied if K is the field of complex numbers. The diagonal entries of the normal form are the eigenvalues of the operator, with the number of times each one occurs being given by its algebraic multiplicity.If the operator is originally given by a square matrix M, then its Jordan normal form is also called the Jordan normal form of M. Any square matrix has a Jordan normal form if the field of coefficients is extended to one containing all the eigenvalues of the matrix. In spite of its name, the normal form for a given M is not entirely unique, as it is a block diagonal matrix formed of Jordan blocks, the order of which is not fixed; it is conventional to group blocks for the same eigenvalue together, but no ordering is imposed among the eigenvalues, nor among the blocks for a given eigenvalue, although the latter could for instance be ordered by weakly decreasing size.The Jordan–Chevalley decomposition is particularly simple with respect to a basis for which the operator takes its Jordan normal form. The diagonal form for diagonalizable matrices, for instance normal matrices, is a special case of the Jordan normal form.The Jordan normal form is named after Camille Jordan.