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In mathematics, an elementary matrix is a square matrix obtained from the application of a single elementary row operation to the identity matrix. The elementary matrices generate the general linear group GL n (F) when F is a field. Left multiplication (pre-multiplication) by an elementary matrix represents elementary row operations, while ...
The following matrices find their main application in statistics and probability theory. Bernoulli matrix — a square matrix with entries +1, −1, with equal probability of each. Centering matrix — a matrix which, when multiplied with a vector, has the same effect as subtracting the mean of the components of the vector from every component.
Find Q minimizing Tr( (Q − M) T (Q − M) ), subject to Q T Q = I. Though written in matrix terms, the objective function is just a quadratic polynomial. We can minimize it in the usual way, by finding where its derivative is zero. For a 3 × 3 matrix, the orthogonality constraint implies six scalar equalities that the entries of Q must satisfy.
Both methods proceed by multiplying the matrix by suitable elementary matrices, which correspond to permuting rows or columns and adding multiples of one row to another row. Singular value decomposition expresses any matrix A as a product UDV ∗, where U and V are unitary matrices and D is a diagonal matrix. An example of a matrix in Jordan ...
A matrix is in reduced row echelon form if it is in row echelon form, with the additional property that the first nonzero entry of each row is equal to and is the only nonzero entry of its column. The reduced row echelon form of a matrix is unique and does not depend on the sequence of elementary row operations used to obtain it.
For matrix-matrix exponentials, there is a distinction between the left exponential Y X and the right exponential X Y, because the multiplication operator for matrix-to-matrix is not commutative. Moreover, If X is normal and non-singular, then X Y and Y X have the same set of eigenvalues. If X is normal and non-singular, Y is normal, and XY ...
An n × n matrix A is diagonalizable if there is a matrix V and a diagonal matrix D such that A = VDV −1. This happens if and only if A has n eigenvectors which constitute a basis for C n . In this case, V can be chosen to be the matrix with the n eigenvectors as columns, and thus a square root of A is
In linear algebra, an invertible matrix is a square matrix which has an inverse. In other words, if some other matrix is multiplied by the invertible matrix, the result can be multiplied by an inverse to undo the operation. An invertible matrix multiplied by its inverse yields the identity matrix. Invertible matrices are the same size as their ...