Square root of a matrix

The square root of a matrix is ​​a concept from linear algebra and generalizes the concept of the square root of a real number. A square root of a square matrix is another matrix that multiplied by itself results in the output matrix. For symmetric positive semidefinite matrices can define a unique square root. For general matrices, however, there must be neither a square root, yet they must, if they exist, be unique.

• 2.1 Definition
• 2.2 Number of existing roots
• 2.3 Geometric Interpretation of Roots
• 2.4 Calculation of a root

Square root of a positive semidefinite matrix

Definition

For a symmetric positive semidefinite matrix is called a symmetric positive semidefinite matrix square root and short root of, if

Applies. The square root of a symmetric positive semidefinite matrix is uniquely determined, it is called with.

Properties and shape

The square root arises from the spectral theorem for symmetric matrices as follows. The spectral theorem provides the existence of an orthogonal matrix and a diagonal matrix, which

And meet with

Again

Result. The roots exist as real numbers, because, thus positive semidefinite and therefore are not all negative.

The fact that the above actually meets the required equality, one sees by

Example

The matrix

Has the eigenvalues ​​with the corresponding orthonormal basis of eigenvectors and. It is therefore

And thus

Square roots of arbitrary matrices

Definition

As a root of a square matrix is any matrix arising multiplied by itself:

You can also find sources in which the root of is if the following applies.

For a root of one also writes

Number of existing roots

As with the square root of a real or complex numbers, the square root of matrices is not necessarily unique. Thus, the identity matrix has infinitely many roots, namely, among others, for each complex number

In addition, as is true in the real or complex numbers: If a root out, then also

Furthermore, there are arrays for which there is no root An example is.

Geometric Interpretation of Roots

Considering the linear transformation matrix that is a mapping between the vector spaces through which a vector is a vector is assigned, then a root is a transformation that is twice must perform in order to convert.

Example:

Is the two-dimensional rotation matrix with the angle

Then is a root of the rotation matrix with the angle (or the angle). The first multiplication is achieved by using a rotation of the half angle and with the second multiplication again.

Calculation of a root

One can easily determine roots of a matrix of size, if is a diagonal matrix or can be converted, at least in a diagonal form (see diagonalization ).

Case 1: diagonal matrix

In the first case, a root is easy to determine by on the diagonal, the root is determined by each element:

For each of the diagonal elements can be arbitrarily choose the sign so that you'll obtain the different solutions.

Since the matrix can also have negative values ​​on the diagonal, the roots can also include complex numbers. However, diagonal matrices with negative diagonal entries may also have real roots; these are then but not themselves diagonal matrices. For example, the following applies:

Case 2: diagonalizable matrix

If the matrix is ​​not a diagonal matrix, they can possibly be converted into diagonal form:

Determining the matrix and the matrix consisting of the eigenvectors of the matrix columns. The matrix is ​​a diagonal matrix with the corresponding eigenvalues ​​of the diagonal.

A number of the matrix is then calculated as follows:

Since is a diagonal matrix, can be rooted calculated as described above. Here, too, it should be noted that the diagonal matrix may include negative eigenvalues ​​, which is the root complex. Since one as in Case 1 for each of the diagonal elements of the matrix can choose the sign arbitrarily here as well, one obtains different solutions here.

Case 3: Non- diagonalizable matrix

Is not diagonalizable, the matrix can be adjusted using the method shown no root calculated. This does not mean that has no root: For example, not diagonalizable, the shear matrix, but has the root

If we allow complex numbers in arithmetic, any matrix can be transformed to Jordan form, even if it is not diagonalizable.

One determines its inverse matrices and with being in the following block diagonal form:

They are Jordan blocks of the form

A root is calculated according to

The root is individually the most of every Jordan block.

If true, the root of a Jordan block is by

Optionally with, wherein the -th derivative of the root function. Explicit results and, with the size of the Jordan block with ( in the diagram), the Subdiagonalen with (the diagonal) and the gamma function are denoted by. The square root of the number is to be used. For thus results, for example

If and is at the same time, the root does not exist out of the Jordan block.

Are all zeros outside the Jordan blocks.

If so, the number has two roots, thus obtained in this way for each Jordan block two different roots. This creates by combining roots, the number of Jordan blocks designated. Proof of the formula by substituting the numbers and exponentiation.

With this method, but you get generally only some and not all square roots of a matrix.