In linear algebra, a diagonal matrix is a square matrix in which all elements outside the main diagonal are zero. Diagonal matrices are therefore determined solely by specifying their main diagonals.
For diagonal matrices, the matrix multiplication and the formation of the inverse can be calculated more easily than with a fully populated matrix. If a linear mapping is represented on a finite-dimensional vector space with the aid of a diagonal matrix, the eigenvalues of the mapping can be read off directly on the basis of the spectral theorem .
the elements with all equal zero, ie diagonal matrix. Often you write for it
is a diagonal matrix.
Special diagonal matrices
The identity matrix is a special case of a diagonal matrix in which all elements of the main diagonal have the value .
The square zero matrix is a special case of a diagonal matrix in which all elements of the main diagonal have the value .
If all the numbers on the main diagonal match in a diagonal matrix, one also speaks of scalar matrices . Scalar matrices are therefore scalar multiples of the identity matrix . The group of the scalar matrices different from the zero matrix is the center of the general linear group .
Properties of diagonal matrices
The respective diagonal matrices form a commutative sub-ring of the ring of square matrices.
The determinant of a diagonal matrix is the product of the entries on the main diagonal:
The adjoint of a diagonal matrix is also a diagonal matrix again.
Multiplication of a matrix from the left by a diagonal matrix corresponds to the multiplication of the rows from with the diagonal entries. The corresponding multiplication from the right corresponds to the multiplication of the columns of with the diagonal entries.
For each diagonal matrix , applies to symmetrically is, therefore applies: .
Calculating the inverse
A diagonal matrix can be inverted if and only if none of the entries is on the main diagonal . The inverse matrix is then calculated as follows: