Search results
Results from the WOW.Com Content Network
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 ...
Let H be a Hadamard matrix of order n.The transpose of H is closely related to its inverse.In fact: = where I n is the n × n identity matrix and H T is the transpose of H.To see that this is true, notice that the rows of H are all orthogonal vectors over the field of real numbers and each have length .
I is the 3 × 3 identity matrix (which is trivially involutory); R is the 3 × 3 identity matrix with a pair of interchanged rows; S is a signature matrix. Any block-diagonal matrices constructed from involutory matrices will also be involutory, as a consequence of the linear independence of the blocks.
In other words, the matrix of the combined transformation A followed by B is simply the product of the individual matrices. When A is an invertible matrix there is a matrix A −1 that represents a transformation that "undoes" A since its composition with A is the identity matrix. In some practical applications, inversion can be computed using ...
Matrix entries are given by the divisor function; entires of the inverse are given by the Möbius function. a ij are 1 if i divides j or if j = 1; otherwise, a ij = 0. A (0, 1)-matrix. Shift matrix: A matrix with ones on the superdiagonal or subdiagonal and zeroes elsewhere. a ij = δ i+1,j or a ij = δ i−1,j
In linear algebra, the adjugate or classical adjoint of a square matrix A, adj(A), is the transpose of its cofactor matrix. [1] [2] It is occasionally known as adjunct matrix, [3] [4] or "adjoint", [5] though that normally refers to a different concept, the adjoint operator which for a matrix is the conjugate transpose.
An idempotent matrix is always diagonalizable. [3] Its eigenvalues are either 0 or 1: if is a non-zero eigenvector of some idempotent matrix and its associated eigenvalue, then = = = = =, which implies {,}.
If A is invertible, the Schur complement of the block A of the matrix M is the q × q matrix defined by /:=. In the case that A or D is singular, substituting a generalized inverse for the inverses on M/A and M/D yields the generalized Schur complement.