Matrix of ones
The matrix is one in mathematics a matrix whose elements are all equal to the number one ( or the unit element of the underlying ring ). A one matrix, which consists of only one row or column, is also called one vector. Each one matrix can be represented as a dyadic product of vectors one. In the matrix ring with matrix addition and the Hadamard product, the one matrix is the identity element. Key figures and powers of Einsmatrizen can be calculated explicitly. The one matrix and one vector should not be confused with the unit matrix and the unit vector.
Definition
Is a ring with one member, the one matrix is defined as
A one matrix consisting of only one row or column is also known as one vector and is denoted by. If the dimension of one matrix from the context clear and there is no possibility of confusion, the indices are omitted and only written. On the basis of unitary matrices, which are often referred to, are also listed by Einsmatrizen.
Examples
Is the field of real numbers and is the number one, so are examples of unity vectors and matrices:
Properties
Algebraic properties
A one matrix can be represented as a dyadic product of vectors one:
The transpose of a matrix one is again a one matrix, ie
The fuel matrix is also the neutral element in the matrix ring, the matrix addition and the Hadamard product are. This applies to all matrices
Rank, determinant, trace
Is now a body, then for the rank of a matrix one
The determinant of a square matrix is then one
The trace of a square matrix one over the real or complex numbers
Eigenvalues
The characteristic polynomial of a real or complex matrix is obtained as the one
The eigenvalues are in accordance
Related are eigenvectors
Products
For the product of two real or complex Einsmatrizen appropriate size
Thus, the -th power of a square matrix for one as calculated
Therefore, the matrix is idempotent, which means
For the Matrixexponential the fuel matrix
The identity matrix of size and the Euler number are.
Programming
In the numerical software package MATLAB One matrix is generated by the function ones ( m, n).