Semidefinite and Second Order Cone Programming Seminar Fall 2001 Lecture 12
... multiplication X ◦ Y = (XY + YX)/2 is a Euclidean Jordan algebra. It is however a Jordan algebra which is not induced by an associative algebra, and in fact it can be shown that it is isomorphic to no Jordan algebra that is a subalgebra of a Jordan algebra induced by an associative algebra. Therefor ...
... multiplication X ◦ Y = (XY + YX)/2 is a Euclidean Jordan algebra. It is however a Jordan algebra which is not induced by an associative algebra, and in fact it can be shown that it is isomorphic to no Jordan algebra that is a subalgebra of a Jordan algebra induced by an associative algebra. Therefor ...
1. Let A = 3 2 −1 1 3 2 4 5 1 . The rank of A is (a) 2 (b) 3 (c) 0 (d) 4 (e
... 14. Let P2 = {a0 +a1 t+a2 t2 } where {a0 , a1 , a2 } range over all real numbers, and let T : P2 → P1 be a linear transformation given by T (a0 +a1 t+a2 t2 ) = a1 +2a2 t. Suppose that B = {1, t, t2 } is a basis of P2 and C = {1, t} is a basis of P1 . (1) Find a matrix A such that [T v]C = A[x]B . (2 ...
... 14. Let P2 = {a0 +a1 t+a2 t2 } where {a0 , a1 , a2 } range over all real numbers, and let T : P2 → P1 be a linear transformation given by T (a0 +a1 t+a2 t2 ) = a1 +2a2 t. Suppose that B = {1, t, t2 } is a basis of P2 and C = {1, t} is a basis of P1 . (1) Find a matrix A such that [T v]C = A[x]B . (2 ...
mathematics 217 notes
... The characteristic polynomial of an n×n matrix A is the polynomial χA (λ) = det(λI −A), a monic polynomial of degree n; a monic polynomial in the variable λ is just a polynomial with leading term λn . Note that similar matrices have the same characteristic polynomial, since det(λI − C −1 AC) = det C ...
... The characteristic polynomial of an n×n matrix A is the polynomial χA (λ) = det(λI −A), a monic polynomial of degree n; a monic polynomial in the variable λ is just a polynomial with leading term λn . Note that similar matrices have the same characteristic polynomial, since det(λI − C −1 AC) = det C ...
![λ1 [ v1 v2 ] and A [ w1 w2 ] = λ2](http://s1.studyres.com/store/data/020256186_1-44523acdcc73497aa300703df377fe57-300x300.png)
The Inverse of a matrix
... vaguely, from high-school algebra. Recall also that the determinant of a matrix is a scalar. Let’s work a specific example: ...
... vaguely, from high-school algebra. Recall also that the determinant of a matrix is a scalar. Let’s work a specific example: ...
MatlabTutorial
... functions. To use these tools, the polynomial should be represented as a vector with the leftmost number being the highest power and the rightmost number being the constant. For example, x² + 2x + 1 would be represented as [1 2 1]. • The roots function gives the roots of the polynomial and polyval e ...
... functions. To use these tools, the polynomial should be represented as a vector with the leftmost number being the highest power and the rightmost number being the constant. For example, x² + 2x + 1 would be represented as [1 2 1]. • The roots function gives the roots of the polynomial and polyval e ...
![1. Let A = 1 −1 1 1 0 −1 2 1 1 . a) [2 marks] Find the](http://s1.studyres.com/store/data/005284378_1-9abef9398f6a7d24059a09f56fe1ac13-300x300.png)
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.