The Linear Algebra Version of the Chain Rule 1
... you can plumb them together only if they fit. After fitting them together the ends in the middle are eliminated, leaving only the outer ends. 2) The matrix product is associative. 3) In general, if AB makes sense, then BA does not. If one restricts to square matrices, i.e. n × n matrices then AB and ...
... you can plumb them together only if they fit. After fitting them together the ends in the middle are eliminated, leaving only the outer ends. 2) The matrix product is associative. 3) In general, if AB makes sense, then BA does not. If one restricts to square matrices, i.e. n × n matrices then AB and ...
Lecture 2 Matrix Operations
... if A is square, and (square) matrix F satisfies F A = I, then • F is called the inverse of A, and is denoted A−1 • the matrix A is called invertible or nonsingular if A doesn’t have an inverse, it’s called singular or noninvertible by definition, A−1A = I; a basic result of linear algebra is that AA ...
... if A is square, and (square) matrix F satisfies F A = I, then • F is called the inverse of A, and is denoted A−1 • the matrix A is called invertible or nonsingular if A doesn’t have an inverse, it’s called singular or noninvertible by definition, A−1A = I; a basic result of linear algebra is that AA ...
3.8 Matrices
... Let A be an m x k matrix and B be a k x n matrix. The product of A and B, denoted by AB, is the m x n matrix with its (i,j)th entry equal to the sum of the products of the corresponding elements from the ith row of A and the jth column of B. In other words, if AB = [cij] then cij = a1j b1j + ...
... Let A be an m x k matrix and B be a k x n matrix. The product of A and B, denoted by AB, is the m x n matrix with its (i,j)th entry equal to the sum of the products of the corresponding elements from the ith row of A and the jth column of B. In other words, if AB = [cij] then cij = a1j b1j + ...
4th 9 weeks
... P.VM.8. Multiply matrices by scalars to produce new matrices, e.g., as when all of the payoffs in a game are doubled. ...
... P.VM.8. Multiply matrices by scalars to produce new matrices, e.g., as when all of the payoffs in a game are doubled. ...
Teacher Notes DOC - TI Education
... Multiplying the encoding matrix by the decoding matrix should result in the identity matrix, since they are inverses. Be sure that students understand that this is true regardless of the order in which the inverse matrices are multiplied. ...
... Multiplying the encoding matrix by the decoding matrix should result in the identity matrix, since they are inverses. Be sure that students understand that this is true regardless of the order in which the inverse matrices are multiplied. ...
Summary of lesson
... Cryptology is the science of developing secret codes and using those codes for encrypting and decrypting data.1 In June of 1929, an article written by Lester S. Hill appeared in the American Mathematical Monthly. This was the first article that linked the fields of algebra and cryptology. 2 Today, g ...
... Cryptology is the science of developing secret codes and using those codes for encrypting and decrypting data.1 In June of 1929, an article written by Lester S. Hill appeared in the American Mathematical Monthly. This was the first article that linked the fields of algebra and cryptology. 2 Today, g ...
notes
... A matrix A is positive definite if x> Ax > 0 for all nonzero x. A positive definite matrix has real and positive eigenvalues, and its leading principal submatrices all have positive determinants. From the definition, it is easy to see that all diagonal elements are positive. To solve the system Ax = ...
... A matrix A is positive definite if x> Ax > 0 for all nonzero x. A positive definite matrix has real and positive eigenvalues, and its leading principal submatrices all have positive determinants. From the definition, it is easy to see that all diagonal elements are positive. To solve the system Ax = ...
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.