Diagonalmatrix

A diagonal matrix is referred to in the mathematical sub-region of linear algebra, a square matrix in which all elements outside of the main diagonal are zero. Diagonal matrices are therefore determined solely by the indication of its main diagonal and one often writes

Votes while all numbers on the main diagonal match, one also speaks of Skalarmatrizen. Skalarmatrizen are therefore scalar multiples of the identity matrix.

• 3.1 Properties of a diagonalizable matrix
• 3.2 diagonalization
• 3.3 Simultaneous diagonalization

Arithmetic operations

Matrix addition, scalar multiplication and matrix multiplication, transposition

The matrix addition, scalar multiplication and matrix multiplication make at diagonal matrices very simple:

A matrix multiplication from the left with a diagonal matrix corresponding to the multiplication of the line of the diagonal entries. The corresponding multiplication from the right corresponds to the multiplication of the columns of the diagonal entries.

Applies for each diagonal matrix, that it is symmetrical, therefore the following applies:.

Calculation of the inverse

A diagonal matrix is invertible if none of the entries on the main diagonal. The inverse matrix is then calculated as follows:

Properties of diagonal matrices

• The respective diagonal matrices form a commutative subring of the ring of square matrices.
• The eigenvalues ​​of a diagonal matrix are the entries on the main diagonal with the canonical unit vectors as eigenvectors.
• The determinant of a diagonal matrix is the product of the entries on the main diagonal of:

Diagonalizability

A square -dimensional matrix is diagonal diagonalisable or similar, where there is a diagonal matrix, to which it is similar, that is to say there exists a invertible matrix, such that, respectively.

For a linear map ( vector space endomorphism ), this means that there is a base, wherein the display matrix is ​​a diagonal matrix.

And are having the desired properties have been found, such that the diagonal entries of, namely, are the eigenvalues ​​of the unit vectors. Continues. So which are also eigenvectors of, respectively corresponding to the eigenvalue.

As should be reversible, is also linearly independent.

In summary, results from the necessary condition that the matrix has linearly independent eigenvectors, the space in which it operates, that has a base of eigenvectors of. However, this condition is also sufficient, because of found eigenvectors with associated eigenvalues ​​can be constructed suitable and very direct.

The problem is thus reduced to finding a sufficient number of linearly independent eigenvectors of.

Properties of a diagonalizable matrix

If a matrix is diagonalizable, then the geometric multiplicity of its eigenvalues ​​is equal to the respective algebraic multiplicities. This means that the dimension of the individual eigenspaces true each consistent with the algebraic multiplicity of the corresponding eigenvalues ​​in the characteristic polynomial of the matrix.

Diagonalization

Is a matrix diagonalisable, there is a diagonal matrix for which the similarity condition is satisfied:

For the diagonalization of this matrix to calculate the diagonal matrix and an associated basis of eigenvectors. This is done in three steps:

Simultaneous diagonalization

Occasionally you may also diagonalize two matrices with the same transformation. If that succeeds, and applies and are there and diagonal matrices,

So the endomorphisms must commute. In fact the converse also holds: commutating two diagonalizable endomorphisms, so they can be diagonalized simultaneously. In quantum mechanics, there are two such operators then a base of common eigenstates.

Example

The showing diagonal matrix

A linear operator has the eigenvalues

With associated eigenspaces / eigenvectors

Special diagonal matrices

• The unit matrix is a special case of a diagonal matrix where all elements of the main diagonal have the value.
• Quadratic zero matrix is a special case of a diagonal matrix where all elements of the main diagonal have the value.
• Normal matrices are diagonalizable. Therefore commutes a complex matrix with its adjoint or a real matrix with its transpose, the matrix is diagonalizable.