Hw #2 pg 109 1-13odd, pg 101 23,25,27,29
... we can multiply the equation by where we get so can be rewritten as which is equivalent to Pg. 101 23. Suppose CA = (the n x n identity matrix). Show that the equation Ax= 0 has only the trivial solution. Explain why A cannot have more columns than rows. We can multiply the vector x to CA = CAx = x ...
... we can multiply the equation by where we get so can be rewritten as which is equivalent to Pg. 101 23. Suppose CA = (the n x n identity matrix). Show that the equation Ax= 0 has only the trivial solution. Explain why A cannot have more columns than rows. We can multiply the vector x to CA = CAx = x ...
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... Jordan algebra. A key instance of the Jordan triple product is the case when x = z (setting x.2 = x.x for notation). Here we get {xyx} = 2(x.y).x − x.2 .y. In a special Jordan algebra this becomes {xyx} = xyx in the associative product. To treat a Jordan algebra as a quadratic Jordan algebra this pr ...
... Jordan algebra. A key instance of the Jordan triple product is the case when x = z (setting x.2 = x.x for notation). Here we get {xyx} = 2(x.y).x − x.2 .y. In a special Jordan algebra this becomes {xyx} = xyx in the associative product. To treat a Jordan algebra as a quadratic Jordan algebra this pr ...
Section 9.5: The Algebra of Matrices
... Big Idea: Matrix arithmetic needs to be defined carefully, and is different than the arithmetic of real numbers. Big Skill: You should be able to multiply matrices, and translate systems of linear equations into matrix equations. 1. Two matrices A = [aij] and B = [bij] are equal if and only if they ...
... Big Idea: Matrix arithmetic needs to be defined carefully, and is different than the arithmetic of real numbers. Big Skill: You should be able to multiply matrices, and translate systems of linear equations into matrix equations. 1. Two matrices A = [aij] and B = [bij] are equal if and only if they ...
Linear Algebra
... Matrix inverse The inverse (A-1) is defined such at A A-1 is I. Not every matrix has an inverse. If no inverse exists, then the matrix is called singular (non invertible) If A is nonsingular, so is A-1 If A, B are nonsingular, then AB is also non singular and (AB) -1 = B -1A -1 (Note revers ...
... Matrix inverse The inverse (A-1) is defined such at A A-1 is I. Not every matrix has an inverse. If no inverse exists, then the matrix is called singular (non invertible) If A is nonsingular, so is A-1 If A, B are nonsingular, then AB is also non singular and (AB) -1 = B -1A -1 (Note revers ...
Proof of the Jordan canonical form
... Theorem 3. There exists a basis of V that is a union of cycles of T . Proof. We will prove this by induction on the dimension of V . If the dimension of V is equal to one, then T is the operator of multiplication by a number, and therefore any non-zero vector of V is an eigenvector. The statement of ...
... Theorem 3. There exists a basis of V that is a union of cycles of T . Proof. We will prove this by induction on the dimension of V . If the dimension of V is equal to one, then T is the operator of multiplication by a number, and therefore any non-zero vector of V is an eigenvector. The statement of ...
Quiz 2 Solutions 1. Let V be the set of all ordered pairs of real
... A(x1 + x2 ) = b + b = 2b 6= b as b is nonzero. To show it is not closed under scalar multiplication, if x1 is a solution and r is real number, then A(rx1 ) = r(Ax1 ) = rb. Note: You only need to show that it fails one of the two closure properties. Also, another argument that it is not a subspace wo ...
... A(x1 + x2 ) = b + b = 2b 6= b as b is nonzero. To show it is not closed under scalar multiplication, if x1 is a solution and r is real number, then A(rx1 ) = r(Ax1 ) = rb. Note: You only need to show that it fails one of the two closure properties. Also, another argument that it is not a subspace wo ...
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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.