MATRICES part 2 3. Linear equations
... A variant of Gaussian elimination called Gauss–Jordan elimination can be used for finding the inverse of a matrix, if it exists. If A is a n by n square matrix, then one can use row reduction to compute its inverse matrix, if it exists. First, the n by n identity matrix is augmented to the right of ...
... A variant of Gaussian elimination called Gauss–Jordan elimination can be used for finding the inverse of a matrix, if it exists. If A is a n by n square matrix, then one can use row reduction to compute its inverse matrix, if it exists. First, the n by n identity matrix is augmented to the right of ...
Solutions
... (Refer to page 27 of the textbook for details). Each of the three equations in the system defines a plane in R3 , with normal vectors given by the rows of the matrix. Recall that two planes are parallel if and only if their normal vectors are collinear. Parallel planes are in fact equal if the equat ...
... (Refer to page 27 of the textbook for details). Each of the three equations in the system defines a plane in R3 , with normal vectors given by the rows of the matrix. Recall that two planes are parallel if and only if their normal vectors are collinear. Parallel planes are in fact equal if the equat ...
NOTES
... Example: consider f : R → R defined by f (x) = 5 for all x. This function is not injective because there is a point in the range (i.e., 5) that gets mapped to by f , starting from a large number of points in the domain. Definition: A function f : X → A is bijective if it is surjective and injective. ...
... Example: consider f : R → R defined by f (x) = 5 for all x. This function is not injective because there is a point in the range (i.e., 5) that gets mapped to by f , starting from a large number of points in the domain. Definition: A function f : X → A is bijective if it is surjective and injective. ...
HOMOLOGY ISOMORPHISMS BETWEEN ALGEBRAIC GROUPS MADE DISCRETE
... E(U ⊗ V, α ⊗ β) = { rs | r ∈ E(U, α) and s ∈ E(V, β) }. Proof. Since tensor product commutes with direct sums, we can assume that U = U (α, r) and V = V (β, s) for r, s ∈ Q. It is then enough to show that for any u ∈ U and v ∈ V there exists a positive integer k such that (α ⊗ β − rs)k (u ⊗ v) = 0. ...
... E(U ⊗ V, α ⊗ β) = { rs | r ∈ E(U, α) and s ∈ E(V, β) }. Proof. Since tensor product commutes with direct sums, we can assume that U = U (α, r) and V = V (β, s) for r, s ∈ Q. It is then enough to show that for any u ∈ U and v ∈ V there exists a positive integer k such that (α ⊗ β − rs)k (u ⊗ v) = 0. ...
spectral theorem for self-adjoint compact operators on Hilbert spaces
... T -stable. Let Z = Y ⊥ . We claim that Z is also T -stable, and that on the Hilbert space Z the (restriction of) T is a compact operator. Indeed, for z ∈ Z and y ∈ Y , we have hT z, yi = hz, T yi = 0 which proves stability easily. And the unit ball in Z is certainly a subset of the unit ball B in X, ...
... T -stable. Let Z = Y ⊥ . We claim that Z is also T -stable, and that on the Hilbert space Z the (restriction of) T is a compact operator. Indeed, for z ∈ Z and y ∈ Y , we have hT z, yi = hz, T yi = 0 which proves stability easily. And the unit ball in Z is certainly a subset of the unit ball B in X, ...
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.