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Lecture Notes for Section 7.2 (Review of Matrices)
... iii. Add any multiple of one row to another row. k. Practice: Compute A-1 again using Gaussian elimination. 10. Matrix Functions: Are matrices with functions for the elements. They can be integrated and differentiated element-by-element… ...
... iii. Add any multiple of one row to another row. k. Practice: Compute A-1 again using Gaussian elimination. 10. Matrix Functions: Are matrices with functions for the elements. They can be integrated and differentiated element-by-element… ...
Transformations with Matrices
... Find the coordinates of the vertices of the image of quadrilateral ABCD with A(–2, 1), B(–1, 4), C(3, 2), and D(4, –2) after a reflection across the line y = x. Write the ordered pairs as a vertex matrix. Then multiply the vertex matrix by the reflection matrix for the line y = x. ...
... Find the coordinates of the vertices of the image of quadrilateral ABCD with A(–2, 1), B(–1, 4), C(3, 2), and D(4, –2) after a reflection across the line y = x. Write the ordered pairs as a vertex matrix. Then multiply the vertex matrix by the reflection matrix for the line y = x. ...
Recitation Notes Spring 16, 21-241: Matrices and Linear Transformations February 9, 2016
... the gray-shaded area can be tiled by exactly one L-shaped tile. Thus we have completely tiled all except the bottom right quadrant. We know that this quadrant with the missing tile (denoted by black shading) can be tiled by assumption since it is a 2n × 2n board with one tile removed. We have tiled ...
... the gray-shaded area can be tiled by exactly one L-shaped tile. Thus we have completely tiled all except the bottom right quadrant. We know that this quadrant with the missing tile (denoted by black shading) can be tiled by assumption since it is a 2n × 2n board with one tile removed. We have tiled ...
Caches and matrix multiply performance; norms
... 1. Split the matrices into 56 × 56 blocks. 2. For each product of 56 × 56 matrices, copy the submatrices into a contiguous block of aligned memory. 3. Further subdivide the 56×56 submatrices into 8×8 submatrices, which are multiplied using a simple fixed-size basic matrix multiply with a few annotat ...
... 1. Split the matrices into 56 × 56 blocks. 2. For each product of 56 × 56 matrices, copy the submatrices into a contiguous block of aligned memory. 3. Further subdivide the 56×56 submatrices into 8×8 submatrices, which are multiplied using a simple fixed-size basic matrix multiply with a few annotat ...
A is square matrix. If
... A square matrix A is called symmetric if A= AT. The entries on the main diagonal may b arbitrary, but “mirror images” of entries across the main diagonal must be equal. ...
... A square matrix A is called symmetric if A= AT. The entries on the main diagonal may b arbitrary, but “mirror images” of entries across the main diagonal must be equal. ...
Problem Set 5
... This problem set will be due Friday, April 26, 2013 at 1 pm in Matti’s mailbox. 1. Let V be an n-dimensional k-vector space, where k is algebraically closed. Then End(V ) can be identified with the vector space of n × n-matrices with k-coefficients, after choosing a basis B. Let Funct(End(V )) denot ...
... This problem set will be due Friday, April 26, 2013 at 1 pm in Matti’s mailbox. 1. Let V be an n-dimensional k-vector space, where k is algebraically closed. Then End(V ) can be identified with the vector space of n × n-matrices with k-coefficients, after choosing a basis B. Let Funct(End(V )) denot ...
PDF
... For the general case, the termination procedure is somewhat more complicated. First recall that a matrix is in echelon form if each row has more leading zeros than the rows above it. A pivot is the leading non-zero entry of some row. We then have • If there is a pivot in the last column, the system ...
... For the general case, the termination procedure is somewhat more complicated. First recall that a matrix is in echelon form if each row has more leading zeros than the rows above it. A pivot is the leading non-zero entry of some row. We then have • If there is a pivot in the last column, the system ...
Generalized Eigenvectors
... This new pair of equations is not completely decoupled, but the equation for qn does not involve pn. We can solve the equation for qn first and substitute the solution into the equation for pn and then solve that. For the equation for qn the solution to the homogeneous equation qn+1 = 2qn is qn = C ...
... This new pair of equations is not completely decoupled, but the equation for qn does not involve pn. We can solve the equation for qn first and substitute the solution into the equation for pn and then solve that. For the equation for qn the solution to the homogeneous equation qn+1 = 2qn is qn = C ...
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.