Uniqueness of the Eigenvalue Inverse Problem for a Class of X-type Matrices with Linear Relations

Constraining a matrix by adding additional conditions to determine the matrix if given eigenpairs is called the eigenvalue inverse problem for matrices. The eigenvalue inverse problem of a matrix can be based on a given combination of different eigenvectors and real numbers to inverse a matrix method, and the method of inversion varies for different types of matrices. In this paper, we investigate the eigenvalue inverse problem for a class of X-type matrices characterized by linear relations, Invert the matrix based on its characteristics, determine eigenvalues and eigenvectors, and finally, prove that the solution of the problem exists and is unique, derive a series of expressions and recursive formulas, and we also check the algorithm’s accuracy correctness by giving different instances.


Introductory
Everything in the world often has a certain order, such as the order of cause and effect, from the cause to push the effect is called a positive problem, and from the effect to find the cause is called the inverse problem [1][2].People have been studying the inverse problem for a long time, there are also inverse problems in mathematics, as the name suggests, the eigenvalue of a matrix to find the eigenvalues and eigenvectors of a matrix, we can understand the eigenvalue of the matrix inverse problem as a known eigenvalues and eigenvectors to determine the matrix [3].
Matrix eigenvalue inverse problems come from a wide range of sources, not only from the discretization of the mathematical inverse problem, but also from many fields such as solid mechanics, particle physics, quantum mechanics, structural design, system parameter identification, automatic control, etc. [4][5], so its study has important theoretical significance and application value.Matrix eigenvalue inverse problems have various approaches due to different conditions or different application backgrounds, such as additive inverse eigenvalue problems [6][7], multiplicative inverse eigenvalue problems, parametric eigenvalue inverse problems [8][9], eigenvalue inverse problems for structured matrices (Jacobi matrix, symmetric banded matrix, Toeplitz matrix, etc.), matrix approximation problem under spectral constraints, and pole configuration problem, etc. [10][11].
Next, we study the eigenvalue inverse problem for a class of X-type matrices characterized by linear relations, given different pairs of eigenpairs to invert the matrices [12][13][14], prove the uniqueness of the solution of the problem, and give the theory, method, and numerical results of solving the problem, and the obtained theory and method have some references value for the later research.
A class of X-type matrices with linear relations behaves as a matrix of the following shape: where, k and l are known constants.
We will discuss the solution and existence uniqueness of a class of eigenvalue inverse problems for matrices of type X with linear relations. ProblemX.
( , , , ) . The essence of problem X is to invert the matrix from the given two sets of feature pairs X .Below we invert the elements of the matrix row by row ( j rows) in several cases , , j j j a b c .

Lemmas and Theorems
Lemma 1.When the expressions for j D and j E are defined by Eqs. ( 4) and (3), it follows that Proof:   According to the method of addition and subtraction, we subtract the two equations ( 7) by multiplying the ends of the equation by j x and the ends of the equation ( 6) by j y , to eliminate j a from the equation. .
, and then according to equation ( 5) we can get ) Taking 2 j  , we have .
We can find the pattern to get equation ( 8)by analogizing the above equation.Because x y can not be zero at the same time, and then combined with ( 6), ( 7), we can get (10). Certificate. Proof: . .
According to the method of addition and subtraction, we multiply the ends of the equation ( 13) by m x and the ends of the equation ( 12) by m y and then subtract the two equations to eliminate them m a from the equation.
Proof: Because of 0 m D  , so from (11) we get (14).Because x y can not be zero at the same time, it can be solved by ( 12),( 13     , the expressions for j D and j E are obtained from Eq. ( 4) and Eq. ( 3), which follows.
1 1 According to the method of addition and subtraction, we subtract the two equations ( 18) by multiplying the ends of the equation by j x and the ends of the equation ( 17) by j y , to eliminate j a from the equation. .
, it follows that ).
, 0, Proof: Let 1 j m   , and from equation ( 8) we get The equation ( 19) is obtained from 16) can be reduced to ) By analogy with the above equation, a pattern can be found to obtain the equation ( 20).
x y cannot be zero at the same time, we can solve j a by combining Eq. ( 17) and ( 18) to obtain Eq.( 22).
Certificate.Lemma 4. When j n  expressions for j D and j E are obtained from Eq. ( 4) and Eq. ( 3), it follows that Proof: , we have 1 . .
According to the method of addition and subtraction, we multiply the ends of the equation (25) by n x and the ends of the equation ( 24) by n y and then subtract the two equations to eliminate n a and get   .
. x y cannot be zero at the same time, we combine it with equations (24), and (25) to obtain equation (27). Certificate.

Xx
x Xy y     .

Summary
In this paper, we study the eigenvalue inverse problem for a class of X-type matrices characterized by linear relations.Under certain conditional constraints, given two known mutually distinct real numbers,   and two nonzero eigenvectors x y  n R , we derive the unique real-order matrix X that corresponds to them, such that ,

Xx
x Xy y     holds, gives the uniqueness theorem for the corresponding solution, and verifies it by an arithmetic example.

2 E
  ; Therefore, if the condition of Theorem 5 above is satisfied, then the problem has a unique solution: