Matrix Differentiation
... Throughout this presentation I have chosen to use a symbolic matrix notation. This choice was not made lightly. I am a strong advocate of index notation, when appropriate. For example, index notation greatly simplifies the presentation and manipulation of differential geometry. As a rule-of-thumb, i ...
... Throughout this presentation I have chosen to use a symbolic matrix notation. This choice was not made lightly. I am a strong advocate of index notation, when appropriate. For example, index notation greatly simplifies the presentation and manipulation of differential geometry. As a rule-of-thumb, i ...
Rotations - FSU Math
... This means the eigenvectors are normalized: their hermitian length is 1 and they are hermitian orthogonal. Now X = BZ and so B is a frame with complex vectors and Z gives the coordinates of the vector X in the frame B. Rotation an angle θ about the z axis is given in the new coordinates by Z → D(θ)Z ...
... This means the eigenvectors are normalized: their hermitian length is 1 and they are hermitian orthogonal. Now X = BZ and so B is a frame with complex vectors and Z gives the coordinates of the vector X in the frame B. Rotation an angle θ about the z axis is given in the new coordinates by Z → D(θ)Z ...
Sections 1.8 and 1.9: Linear Transformations Definitions: 1
... = A(cu) + A(dv) (distributing matrix multiplication) ...
... = A(cu) + A(dv) (distributing matrix multiplication) ...
4.3 - shilepsky.net
... find a vector x in Rn such that TA(x) = w. That is, Ax = w. This is the same as saying Ax = w is consistent for all nx1 vectors w. We have already shown this is equivalent to A being invertible. We show part c) is equivalent to A is invertible by showing the following: TA is 1-1 if and only if Ax = ...
... find a vector x in Rn such that TA(x) = w. That is, Ax = w. This is the same as saying Ax = w is consistent for all nx1 vectors w. We have already shown this is equivalent to A being invertible. We show part c) is equivalent to A is invertible by showing the following: TA is 1-1 if and only if Ax = ...
Leslie and Lefkovitch matrix methods
... There is one last variable to calculate. Vx , or the reproductive value, measures the relative importance of different age classes to the future reproduction of a population. If many die before reaching the age at which reproduction begins, those organisms are less important to future growth than ar ...
... There is one last variable to calculate. Vx , or the reproductive value, measures the relative importance of different age classes to the future reproduction of a population. If many die before reaching the age at which reproduction begins, those organisms are less important to future growth than ar ...
Solving Systems of Equations
... are matrices, the number of columns in A must match the number of rows in B. If this condition is not met, then the product AB does not exist. Check your text for a summary and examples. If this condition is met, we then form a linear combination of the entries in the top row of A and the entries in ...
... are matrices, the number of columns in A must match the number of rows in B. If this condition is not met, then the product AB does not exist. Check your text for a summary and examples. If this condition is met, we then form a linear combination of the entries in the top row of A and the entries in ...
Cayley-Hamilton theorem over a Field
... original F[X] module. The next operations will alter this, but we will always have *valid equations, when you stick in the vector multipliers e_1 e_2 etc into such an array as above. Our goal is to create a diagonal matrix from W. There are operations or steps of two types. Each type of step involve ...
... original F[X] module. The next operations will alter this, but we will always have *valid equations, when you stick in the vector multipliers e_1 e_2 etc into such an array as above. Our goal is to create a diagonal matrix from W. There are operations or steps of two types. Each type of step involve ...
Lecture 8: Examples of linear transformations
... In general, shears are transformation in the plane with the property that there is a vector w ~ such that T (w) ~ =w ~ and T (~x) − ~x is a multiple of w ~ for all ~x. Shear transformations are invertible, and are important in general because they are examples which can not be diagonalized. ...
... In general, shears are transformation in the plane with the property that there is a vector w ~ such that T (w) ~ =w ~ and T (~x) − ~x is a multiple of w ~ for all ~x. Shear transformations are invertible, and are important in general because they are examples which can not be diagonalized. ...
notes II
... The nullspace of A The nullspace of A is defined by solutions to A.x = 0. If we transform A.x = 0 to U.x = 0 by a set of row operations then it is obvious that solutions to U.x = 0 must be the same as those to A.x = 0 and so the nullspace of A is the same as the nullspace of U. It has dimension n – ...
... The nullspace of A The nullspace of A is defined by solutions to A.x = 0. If we transform A.x = 0 to U.x = 0 by a set of row operations then it is obvious that solutions to U.x = 0 must be the same as those to A.x = 0 and so the nullspace of A is the same as the nullspace of U. It has dimension n – ...
poster
... We therefore want generalized DFTs that show us similar symmetry-invariant structure. We can write these symmetries abstractly as groups and define these new DFTs using tools from abstract algebra: ...
... We therefore want generalized DFTs that show us similar symmetry-invariant structure. We can write these symmetries abstractly as groups and define these new DFTs using tools from abstract algebra: ...
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.