Matrices for which the squared equals the original
... The identity matrix is defined as any square matrix with the property that all entries along the main diagonal are the value 1, the others being zero. This special matrix is denoted by In and has the unique property such that for any m * n matrix A, Im * A = A = A * In, where Ix is a square matrix o ...
... The identity matrix is defined as any square matrix with the property that all entries along the main diagonal are the value 1, the others being zero. This special matrix is denoted by In and has the unique property such that for any m * n matrix A, Im * A = A = A * In, where Ix is a square matrix o ...
Full text
... There are numerous applications of linear operators and matrices that give rise to tridiagonal matrices. Such applications occur naturally in mathematics, physics, and chemistry, e.g., eigenvalue problems, quantum optics, magnetohydrodynamics and quantum mechanics. It is convenient to have theoretic ...
... There are numerous applications of linear operators and matrices that give rise to tridiagonal matrices. Such applications occur naturally in mathematics, physics, and chemistry, e.g., eigenvalue problems, quantum optics, magnetohydrodynamics and quantum mechanics. It is convenient to have theoretic ...
Linear transformations and matrices Math 130 Linear Algebra
... Aha! You see it! If you take the elements in the ith row from A and multiply them in order with elements in the column of v, then add those n products together, you get the ith element of T (v). With this as our definition of multiplication of an m × n matrix by a n × 1 column vector, we have Av = T ...
... Aha! You see it! If you take the elements in the ith row from A and multiply them in order with elements in the column of v, then add those n products together, you get the ith element of T (v). With this as our definition of multiplication of an m × n matrix by a n × 1 column vector, we have Av = T ...
Theorem: (Fisher`s Inequality, 1940) If a (v,b,r,k,λ) – BIBD exists with
... then combined to obtain the colored picture. ...
... then combined to obtain the colored picture. ...
Module 4 : Solving Linear Algebraic Equations Section 3 : Direct
... systems. When number of equations is significantly large ( ), even the Gaussian elimination and related methods can turn out to be computationally expensive and we have to look for alternative schemes that can solve (LAE) in smaller number of steps. When matrices have some simple structure (few non- ...
... systems. When number of equations is significantly large ( ), even the Gaussian elimination and related methods can turn out to be computationally expensive and we have to look for alternative schemes that can solve (LAE) in smaller number of steps. When matrices have some simple structure (few non- ...
Sections 3.4-3.6
... The determinant of square matrix A is a number |A| associated with the matrix. The determinant of a two-by-two matrix is the product of the diagonal elements minus the product of the off-diagonal elements: a11 a12 a11a22 a12 a21 . a21 a22 Before calculating the determinant of an arbitrary n n ...
... The determinant of square matrix A is a number |A| associated with the matrix. The determinant of a two-by-two matrix is the product of the diagonal elements minus the product of the off-diagonal elements: a11 a12 a11a22 a12 a21 . a21 a22 Before calculating the determinant of an arbitrary n n ...
03.Preliminaries
... guarantee that if a vector of the basis, say aj, is replaced by another vector, say ap, then the new set of vectors still forms a basis? ...
... guarantee that if a vector of the basis, say aj, is replaced by another vector, say ap, then the new set of vectors still forms a basis? ...
Some Computations in Support of Maeda`s Conjecture
... integers s1 , . . . , sr ∈ Fp . By the Chinese remainder theorem, there is a unique polynomial v(x) of degree less than n that reduces to si modulo qi for all i. This polynomial has an interesting property: modulo f , v(x)p ≡ v(x), because, modulo each of the qi , we have v(x)p ≡ spi = si ≡ v(x), wi ...
... integers s1 , . . . , sr ∈ Fp . By the Chinese remainder theorem, there is a unique polynomial v(x) of degree less than n that reduces to si modulo qi for all i. This polynomial has an interesting property: modulo f , v(x)p ≡ v(x), because, modulo each of the qi , we have v(x)p ≡ spi = si ≡ v(x), wi ...
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.