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Multiequilibria analysis for a class of collective decision
... Consider the system (7) (or (6)) where each nonlinear function ψi (xi ) satisfies the properties (A.1)÷(A.4). Let us start by recalling what is known for this system when we vary the parameter π. By construction, the origin is always an equilibrium point for (7) (or (6)). When π < 1, x = 0 is the on ...
... Consider the system (7) (or (6)) where each nonlinear function ψi (xi ) satisfies the properties (A.1)÷(A.4). Let us start by recalling what is known for this system when we vary the parameter π. By construction, the origin is always an equilibrium point for (7) (or (6)). When π < 1, x = 0 is the on ...
Uniqueness of the row reduced echelon form.
... row operations on a m × n matrix A are 1. Multiplying a row of A by a nonzero scalar. 2. Adding a scalar multiple of one row to anther row. 3. Interchanging two rows of A. In terms of the corresponding system of equations these operations are the same as multiplying one of the equations by a nonzero ...
... row operations on a m × n matrix A are 1. Multiplying a row of A by a nonzero scalar. 2. Adding a scalar multiple of one row to anther row. 3. Interchanging two rows of A. In terms of the corresponding system of equations these operations are the same as multiplying one of the equations by a nonzero ...
NONCOMMUTATIVE JORDAN ALGEBRAS OF
... Let a = x"~3 in (5): xxn~l=x"~2x2 = x"~1x=xn. We need to prove x"~ax" = x" for a = l, • • • , n — 1, and prove this by proving xn~"x" =xn =x"x"~a by induction on a. This has been proved for a = l, and we assume xn~ffx^ = xn=x^xn^ for all fiSa as follo ...
... Let a = x"~3 in (5): xxn~l=x"~2x2 = x"~1x=xn. We need to prove x"~ax" = x" for a = l, • • • , n — 1, and prove this by proving xn~"x" =xn =x"x"~a by induction on a. This has been proved for a = l, and we assume xn~ffx^ = xn=x^xn^ for all fiSa
Mathcad Professional
... An important point about the range variable statement for real valued variables: the ".." symbol is achieved by typying the semicolon key. Graphing functions: under "Insert" (see menu at top) is the choice "Graph", followed by "X-Y plot". This generates a graph which can be used with the range varia ...
... An important point about the range variable statement for real valued variables: the ".." symbol is achieved by typying the semicolon key. Graphing functions: under "Insert" (see menu at top) is the choice "Graph", followed by "X-Y plot". This generates a graph which can be used with the range varia ...
Definition
... An identity matrix is a square matrix with the digit 1 as the elements along its main diagonal ( sloping downwards from left to right ) and all the other elements being zeros. ...
... An identity matrix is a square matrix with the digit 1 as the elements along its main diagonal ( sloping downwards from left to right ) and all the other elements being zeros. ...
Introduction to Matrix Algebra
... (1− λ )a1 +ρa2 = 0 (1− λ )a2 + ρ a1 = 0 Now take the largest eigenvalue, l = 1 + r, and substitute. This gives ρ( a2 − a1 ) = 0 ρ( a1 − a2 ) = 0 Thus, all we know is that a 1 = a 2 . If we let a 1 = 10, then a 2 = 10; and if we let a 1 = -.023, then a 2 = -.023. This is what was meant above when it ...
... (1− λ )a1 +ρa2 = 0 (1− λ )a2 + ρ a1 = 0 Now take the largest eigenvalue, l = 1 + r, and substitute. This gives ρ( a2 − a1 ) = 0 ρ( a1 − a2 ) = 0 Thus, all we know is that a 1 = a 2 . If we let a 1 = 10, then a 2 = 10; and if we let a 1 = -.023, then a 2 = -.023. This is what was meant above when it ...
Fall 2012 Midterm Answers.
... you understand notation. You are told that C = AB, so {Cx | x ∈ Rp } = {ABx | x ∈ Rp }. A vector ABx is certainly a (right) multiple of A. (24) The linear span of the vectors (4, 0, 0, 1), (0, 2, 0, −1) and (4, 3, 2, 1) is the 3-plane 2x1 − 2x2 + 3x3 − 4x4 = 0 in R4 . False. [66,7,27] Comments: The ...
... you understand notation. You are told that C = AB, so {Cx | x ∈ Rp } = {ABx | x ∈ Rp }. A vector ABx is certainly a (right) multiple of A. (24) The linear span of the vectors (4, 0, 0, 1), (0, 2, 0, −1) and (4, 3, 2, 1) is the 3-plane 2x1 − 2x2 + 3x3 − 4x4 = 0 in R4 . False. [66,7,27] Comments: The ...
Solutions to Math 51 Second Exam — February 18, 2016
... • det(A) = 0. • The first and second columns of A are the same. • Every column of A has only one non-zero entry in it. • All the diagonal entries of A are zero: a11 = a22 = a33 = a44 = a55 = 0. The grading scheme for part (b) was that you got .75 points for a correct answer and -.75 points for an in ...
... • det(A) = 0. • The first and second columns of A are the same. • Every column of A has only one non-zero entry in it. • All the diagonal entries of A are zero: a11 = a22 = a33 = a44 = a55 = 0. The grading scheme for part (b) was that you got .75 points for a correct answer and -.75 points for an in ...
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.