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Kronecher product is an important matrix product, which plays an important role in the study of matrix theory and is a basic mathematical tool. This paper introduces the basic properties of matrix direct product and uses it to solve linear matrix equations and matrix differential equations.

Definition and properties of direct product

  • Definition 8.1: Suppose the matrix:

The following block matrix is called:


forwithThe direct product or the Kronecher product.

visibleisMatrix.

The direct product of a matrix has the following properties:

  • 1,Is a constant, then:

  • 2,forWith matrix, then:


  • 3, :

  • 4. :

  • 5, a:..., then:

  • 6, a:

allreversible,alsoreversibleAnd:


  • 7,Set,Are allThe unitary matrix,Is alsoThe unitary matrix.

  • Eight,Of all theThe eigenvalueis....Of all theThe eigenvalueis...,theEnsemble eigenvalueIs this:


  • 9,.,.

  • 10,The eigenvalue of is....The eigenvalue of is...Is:


The eigenvalues of are:


  • 11,The eigenvalue of is....The eigenvalue of is...,And the eigenvalues of+.

  • setistheThe feature vectors.istheThe feature vectors,istheThe feature vectors.

  • set, then:


  • set.Is:

Applications of direct products

This section discusses the application of direct product in solving linear matrix equations.

straighten

  • Definition 8.2: a matrixSaid,Dimensional column vector:

forThe straight.

Straightening has the following properties:

  1. set.withIs a constant, then:

  1. setIs:

  • Theorem 8.1 Suppose:

Is:




Linear matrix equations

  • The solutions of several types of equations are discussed below: let

Solution of Lyapunov matrix equation:

The solution straightens both sides of the matrix:


Since the matrix equation is equivalent to the linear equations obtained, the necessary and sufficient conditions for obtaining the matrix equations to have solutions are:



The necessary and sufficient conditions for a unique solution are:


  • setAnd,The eigenvalues of are all real numbers, and the matrix equation is proved:

There’s a unique solution.

  • In the set

Solving the initial value problem of matrix differential equations:


The solution straightens both sides of the matrix:


This is a system of homogeneous linear differential equations with constant coefficients, whose solution is:



By:



So: