Is a zero matrix diagonalizable

Diagonal matrix

 

As Diagonal matrix In the mathematical sub-area of ​​linear algebra, one denotes a square matrix in which all elements outside the main diagonal are zero. Diagonal matrices are therefore determined solely by specifying their main diagonal and one writes frequently

.

All the numbers are correct on the main diagonal, one also speaks of Scalar matrices. So scalar matrices are scalar multiples of the identity matrix.

Arithmetic operations

Matrix addition, scalar multiplication and matrix multiplication, transposition

The matrix addition, scalar multiplication and matrix multiplication are very simple with diagonal matrices:

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

For every diagonal matrix it holds that it is symmetrical, hence: .

Calculation of the inverse

A diagonal matrix can be inverted if and only if none of the entries are on the main diagonal is. The inverse matrix is ​​then calculated as follows:

Properties of diagonal matrices

  • The respective diagonal matrices form a commutative sub-ring of the ring of the 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:

Examples

The diagonal matrix

owns the eigenvalues

with the associated eigenvectors

.

The diagonal matrix

owns the eigenvalues

and the eigenvectors

.

Diagonalisability

A square one -dimensional matrix called diagonalizable or diagonal likeif there is a diagonal matrix to which it is similar, that is, there is a regular matrixso that applies , or. .

For a linear map (Vector space endomorphism) it means that a base exists in which the display matrix is a diagonal matrix.

Be and found with the desired properties, then the diagonal entries of , namely , Eigenvalues ​​of to the unit vectors are. Furthermore is . The are also eigenvectors of , in each case at the eigenvalue .

There should be invertible is also linearly independent.

In summary, this results in the necessary condition that the matrix has linearly independent eigenvectors, the space on which it operates has a basis of eigenvectors of owns. However, this condition is also sufficient because it is off found eigenvectors of with the associated eigenvalues, suitable and construct very directly.

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

Properties of a diagonalizable matrix

If a matrix can be diagonalized, the geometric multiplicity of its eigenvalues ​​is equal to the respective algebraic multiplicity. This means that the dimension of the individual eigenspaces corresponds in each case to the algebraic multiplicity of the corresponding eigenvalues ​​in the characteristic polynomial of the matrix.

Diagonalization

Is a matrix Diagonalizable, there is a diagonal matrix , for which the similarity condition is fulfilled:

The diagonal matrix is ​​calculated to diagonalize this matrix and an associated basis of eigenvectors. This is done in three steps:

  1. It will be the eigenvalues the matrix certainly.
  2. It will be the eigenspaces to all eigenvalues calculated, i.e. solved the following system of equations:
  3. Now is the diagonal shape the matrix regarding the base :

Simultaneous diagonalization

Occasionally you also want two matrices with the same transformation diagonalize. If that succeeds, then applies and and since and Are diagonal matrices,

.

So the endomorphisms have to commute with one another. In fact, the reverse also applies: if two diagonalizable endomorphisms commutate, they can be diagonalized simultaneously. In quantum mechanics there is a basis of common eigenstates for two such operators.

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 to have.
  • The square zero matrix is ​​a special case of a diagonal matrix in which all elements of the main diagonal have the value to have.
  • Normal matrices can be diagonalized. If a complex matrix commutates with its adjoint or a real matrix with its transpose, then the matrix can be diagonalized. In the real case, it must be assumed that the characteristic polynomial of the matrix over the real numbers breaks down into a product of linear factors. The corresponding condition is automatically fulfilled for the complex numbers due to the fundamental theorem of algebra.

See also

Based on an article in: Wikipedia.de
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Date of the last change: Jena, the: 27.12. 2020