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).

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