Midterm 2
... C A B where cij aij bij , 1 i n, 1 j m . Multiplication: Let A be an n-by-m matrix, and B be an m-by-p matrix. Then the multiplication of A and B is an n-by-p matrix m ...
... C A B where cij aij bij , 1 i n, 1 j m . Multiplication: Let A be an n-by-m matrix, and B be an m-by-p matrix. Then the multiplication of A and B is an n-by-p matrix m ...
PreCalculus - TeacherWeb
... *Elementary Row operations (see key concept on page 366): Each of the following row operations produces an equivalent augmented matrix. Interchange any 2 rows Multiply one row by a nonzero real number Add a multiple of one row to another row *See page 367 for comparison of Gaussian Elimination ...
... *Elementary Row operations (see key concept on page 366): Each of the following row operations produces an equivalent augmented matrix. Interchange any 2 rows Multiply one row by a nonzero real number Add a multiple of one row to another row *See page 367 for comparison of Gaussian Elimination ...
Classwork 84H TEACHER NOTES Perform Reflections Using Line
... Go to CW and go through “Common lines of reflection in the coordinate plane” (#1 – 4) Using matrices to reflect figures We can also use matrices to reflect figures… Let’s use segment AB in the coordinate plane to be A(1, 3) and B(2, 4). Let’s see what happens when we multiply the matrix for segmen ...
... Go to CW and go through “Common lines of reflection in the coordinate plane” (#1 – 4) Using matrices to reflect figures We can also use matrices to reflect figures… Let’s use segment AB in the coordinate plane to be A(1, 3) and B(2, 4). Let’s see what happens when we multiply the matrix for segmen ...
Matrix manipulations
... Many “rookie mistakes” in linear algebra involve forgetting ways in which matrices differ from scalars: • Not all matrices are square. • Not all matrices are invertible (even nonzero matrices can be singular). • Matrix multiplication is associative, but not commutative. Don’t forget these facts! In ...
... Many “rookie mistakes” in linear algebra involve forgetting ways in which matrices differ from scalars: • Not all matrices are square. • Not all matrices are invertible (even nonzero matrices can be singular). • Matrix multiplication is associative, but not commutative. Don’t forget these facts! In ...
DOC - math for college
... What are some of the rules of binary matrix operations? Commutative law of addition If [ A] and [B] are m n matrices, then [ A] [ B] [ B] [ A] Associative law of addition If [A], [B] and [C] are all m n matrices, then [ A] [ B] [C] [ A] [ B] [C] Associative law of multiplicatio ...
... What are some of the rules of binary matrix operations? Commutative law of addition If [ A] and [B] are m n matrices, then [ A] [ B] [ B] [ A] Associative law of addition If [A], [B] and [C] are all m n matrices, then [ A] [ B] [C] [ A] [ B] [C] Associative law of multiplicatio ...
SheilKumar
... z0 is the smallest eigenvector of D-1/2(D-W)D-1/2, and all other eigenvectors are perpendicular to it. z1, the second smallest eigenvector has the property ...
... z0 is the smallest eigenvector of D-1/2(D-W)D-1/2, and all other eigenvectors are perpendicular to it. z1, the second smallest eigenvector has the property ...
Iterative methods to solve linear systems, steepest descent
... While computing the exact solution by hand would be a tedious task, it is a simple matter to find an approximate solution. Roughly speaking, we expect the behavior of our system to be governed by the “large” diagonal entries of the matrix A, so if we just pretend that all off-diagonal entries are zero ...
... While computing the exact solution by hand would be a tedious task, it is a simple matter to find an approximate solution. Roughly speaking, we expect the behavior of our system to be governed by the “large” diagonal entries of the matrix A, so if we just pretend that all off-diagonal entries are zero ...
CSCE 590E Spring 2007
... If V and W are unit vectors, the dot product is 1 for parallel vectors pointing in the same direction, -1 for opposite ...
... If V and W are unit vectors, the dot product is 1 for parallel vectors pointing in the same direction, -1 for opposite ...
Algebra 3 – Chapter 10 – Matrices
... If an m n has the same number of rows as columns, that is if m = n, the matrix is referred to as a square matrix. Two m by n matrices A and B are said to be equal, written as A = B provided that each entry bij in A is equal to the corresponding entry bij in B. The Identity Matrix is a square matri ...
... If an m n has the same number of rows as columns, that is if m = n, the matrix is referred to as a square matrix. Two m by n matrices A and B are said to be equal, written as A = B provided that each entry bij in A is equal to the corresponding entry bij in B. The Identity Matrix is a square matri ...
10.3 POWER METHOD FOR APPROXIMATING EIGENVALUES
... In this section you have seen the use of the power method to approximate the dominant eigenvalue of a matrix. This method can be modified to approximate other eigenvalues through use of a procedure called deflation. Moreover, the power method is only one of several techniques that can be used to app ...
... In this section you have seen the use of the power method to approximate the dominant eigenvalue of a matrix. This method can be modified to approximate other eigenvalues through use of a procedure called deflation. Moreover, the power method is only one of several techniques that can be used to app ...
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.