Zero matrix

A zero matrix in linear algebra a real or complex matrix whose entries are all equal to the number zero. Generally means a matrix on a body or ring zero matrix, when all matrix elements correspond to the identity element of the addition in the body or ring. The zero matrix represents the zero mapping between finite dimensional vector spaces and is even in the neutral element in the vector space or a ring of matrices. The most important characteristics of a zero matrix as determinant, trace and rank, are each zero. The transpose, adjoint or complementary matrix of a zero matrix is again a zero matrix.

Definition

Is a ring having zero element, it is the zero matrix is defined as

The entries of a zero matrix are therefore all equal to the zero element of the ring. The zero matrix is ​​, if their dimension is irrelevant and there is no possibility of confusion, just quoted also by or. A matrix without contents, in which therefore the number of lines or columns is equal to zero is called " blank matrix ". Such a matrix is ​​always a zero matrix and, if square, at the same time unit matrix.

Examples

Is the field of real numbers and the numeral zero, are examples of zero matrices:

Properties

Neutrality

Between two finite dimensional vector spaces over the same body is the zero matrix represents the zero picture, so the linear map that maps all vectors to the zero vector. If the zero vector of the target space, then for all vectors

In the vector space of matrices is the zero matrix itself the zero vector with respect to the matrix addition is, which means it applies to all matrices

In the matrix ring is the zero matrix the zero element and the unit matrix corresponds to the identity element. With respect to the matrix multiplication, the zero matrix acts as an absorbing element, because valid for all matrices

A zero matrix therefore has no ( multiplicative ) inverse, as the product of the zero- matrix using any matrix may not give the identity matrix. The ring of square matrices is not zero divisors, because of not necessarily follow or.

Parameters

For the determinant and the trace of a square zero matrix

Also applies to the rank of a zero matrix over a field

With zero matrices are the only matrices with rank zero. The characteristic polynomial of a quadratic matrix zero over a body in the form of

Thus the only eigenvalue and the corresponding eigenspace is the whole space. A square zero matrix over the real or complex numbers is both positive semidefinite and negative semidefinite.

Pictures

Each zero matrix may be represented as the dyadic product of two zero vectors of appropriate length, thus

The transposed matrix, adjoint matrix or a zero matrix of complementary matrix is again a zero matrix in which only the dimensions are exchanged:

The Matrixexponential a real or complex square zero matrix is ​​the identity matrix of the same size, short