Parallel eigendecomposition method of Hermitian matrix for electromagnetic reconnaissance system

Abstract

This invention relates to the field of electromagnetic reconnaissance technology, specifically disclosing a parallel eigenvalue decomposition method for Hermitian matrices in electromagnetic reconnaissance systems. The method constructs a 2N-order real symmetric matrix S based on the real and imaginary parts of an N-order Hermitian matrix of the electromagnetic reconnaissance system. Then, based on a scheduling rule matrix T, it performs M iterative calculations on the 2N-order real symmetric matrix S and the eigenvector matrix R to obtain the corresponding real symmetric matrix S' and eigenvector matrix R'. Finally, based on the elements of matrices S' and R', the eigenvalues ​​of the N-order Hermitian matrix and the corresponding eigenvectors are obtained. This invention transforms complex matrices into real matrices for processing and designs a scheduling rule matrix T for iterative calculation, achieving high decomposition accuracy. The same processing method can be applied to eigenvalue decomposition for Hermitian matrices of odd or even dimensions, making it applicable to any processing system that supports parallel computing, thus having a wide range of applications.

Description

Technical Field

[0001] This invention relates to the field of electromagnetic reconnaissance technology, and in particular to a parallel eigenvalue decomposition method for Hermitian matrices used in electromagnetic reconnaissance systems. Background Technology

[0002] Hermitian matrices are a class of key matrices in the complex field, widely used in mathematics, physics, and engineering. In engineering, they are commonly used as column vectors for complex signals. Its conjugate transpose Multiplication yields the covariance matrix That is, the Hermitian matrix. For the covariance matrix... Eigenvalue decomposition yields multiple eigenvalues ​​and their corresponding eigenvectors, which have specific physical meanings in different fields. For example, in structural mechanics, eigenvalues ​​represent natural frequencies, and eigenvectors represent mode shapes; in electromagnetic reconnaissance systems, large eigenvalues ​​represent signal power, and small eigenvalues ​​represent noise power. The commonly used eigenvalue decomposition algorithm in engineering is the QR method, but its implementation is complex and difficult to parallelize. The Jacobi eigenvalue decomposition method, on the other hand, has advantages such as high accuracy, good parallelism, and strong stability, but it is only applicable to real symmetric matrices.

[0003] The invention patent with patent number 201810120721.4, entitled "Method for Implementing Hermitian Matrix Decomposition Using FPGA", provides an eigenvalue decomposition method for Hermitian matrices implemented on FPGA. However, different processing methods are required for Hermitian matrices with odd and even dimensions, and there is no unified algorithm applicable to the eigenvalue decomposition of Hermitian matrices with arbitrary dimensions. Summary of the Invention

[0004] This invention provides a parallel eigenvalue decomposition method for Hermitian matrices in electromagnetic reconnaissance systems. The technical problem it solves is that current Jacobi parallel eigenvalue decomposition methods are mainly for real matrices, or require separate processing for odd-dimensional and even-dimensional Hermitian matrices. There is no unified algorithm applicable to eigenvalue decomposition of Hermitian matrices of arbitrary dimensions.

[0005] To address the above technical problems, this invention provides a parallel eigenvalue decomposition method for the Hermitian matrix in electromagnetic reconnaissance systems, comprising the following steps:

[0006] S1. Obtain the N-channel complex signal vectors of the electromagnetic reconnaissance system and construct the N-order Hermitian matrix. ;

[0007] S2. Extract the Nth-order Hermitian matrix. We obtain an N-order real matrix A and an N-order imaginary matrix B from the real and imaginary parts of the matrix; and construct a 2N-order real symmetric matrix S based on matrices A and B.

[0008] S3, Generation The scheduling rule matrix T is such that each row of matrix T consists of 2N elements from 1 to 2N arranged in different orders, and the elements in the first column are all 1;

[0009] S4. Based on the scheduling rule matrix T, perform M iterations on the 2N-order real symmetric matrix S and the eigenvector matrix R to obtain the corresponding real symmetric matrix S' and eigenvector matrix R'.

[0010] S5. Based on the elements of matrices S' and R', obtain the N-order Hermitian matrix. The eigenvalues ​​and the corresponding eigenvectors of each eigenvalue.

[0011] Further, in step S2, matrix S is constructed as follows: .

[0012] Furthermore, in step S3, the arrangement rule for rows 1 to N of matrix T is as follows:

[0013] All elements in the first column are 1;

[0014] Starting from the first digit of “2, 4, 6, ..., 2N, 2N-1, 2N-3, 2N-5, ..., 5, 3”, slide the window to the right with a length of N to obtain N sets of data, and fill them sequentially into the first row to the Nth row of the even-numbered column of matrix T.

[0015] Starting from the last character of “3, 5, ..., 2N-1, 2N, 2N-2, ..., 4”, slide the window to the left with a length of N to obtain N-1 sets of data, and fill them sequentially into the even-numbered columns of the N+1th row to the 2N-1th row of matrix T;

[0016] Starting from the last position of “2N-2, 2N-4, ..., 4, 2, 3, 5, ..., 2N-3, 2N-1”, slide the window to the left with a length of N-1 to obtain N-1 sets of data, and fill them sequentially into the remaining odd columns from the first row to the Nth row of matrix T;

[0017] Starting from the first character of “2N-3, 2N-5, ..., 3, 2, 4, ..., 2N-2”, slide the window to the right with a length of N-1 to obtain N-1 sets of data, and fill them sequentially into the remaining odd columns from row N+1 to row 2N-1 of matrix T.

[0018] Furthermore, step S4 specifically includes the following steps:

[0019] S41. Assign the 2N×2N identity matrix I to matrix R and initialize the iteration count to 0;

[0020] S42. Determine if the number of iterations is less than M. If so, increment the number of iterations by 1, initialize the number of iterations to 0, and proceed to the next step. Otherwise, output the current matrices S and R as the final real symmetric matrix S' and eigenvector matrix R'.

[0021] S43. Determine if the number of iterations is less than 2N-1. If not, return to the previous step. If yes, execute steps S44 to S46.

[0022] S44. Assign the identity matrix I to matrix W;

[0023] S45. Update the element value of the corresponding row and column index in matrix W according to the row and column index pair of the current row of the scheduling rule matrix T and the 2N-order real symmetric matrix S.

[0024] S46. Update matrices S and R based on the updated matrix W, and then return to step S43.

[0025] Furthermore, step S45 specifically includes the following steps:

[0026] S451. Calculate the cosine value c(j) and sine value s(j) of the rotation angle corresponding to the current iteration step based on matrix S and matrix T. ;

[0027] S452, based on , , Update matrix W OK List, OK List, OK List, OK The elements of the column.

[0028] Further, in step S451, , Calculated by the following formula:

[0029] ,

[0030] ,

[0031] ,

[0032] ,

[0033] in, This represents the cotangent value at twice the rotation angle. Representation matrix middle OK The element value, Represents a symbolic function.

[0034] Further, in step S46, the matrix is ​​updated. sum matrix The superscript H indicates the conjugate transpose.

[0035] Furthermore, step S5 specifically includes the following steps:

[0036] S51. Sort the diagonal elements of the real symmetric matrix S' in ascending order and record the sorted odd index sequence: index(1), index(3), ..., index(2N-1);

[0037] S52. The elements corresponding to the odd indices 1, 3, ..., 2N-1 of the diagonal elements of matrix S' after ascending sorting of its diagonal elements are the matrix itself. The eigenvalues ​​arranged in ascending order;

[0038] S53. Arrange the matrix according to the sorted odd index sequence. The corresponding columns are extracted to form matrix UV. The first N rows of matrix UV are taken to form matrix U, and the last N rows are taken to form matrix V. Then the matrix... Each column is an eigenvector corresponding to the eigenvalues ​​arranged in ascending order.

[0039] The parallel eigenvalue decomposition method for the Hermitian matrix of electromagnetic reconnaissance systems provided by this invention is based on the N-order Hermitian matrix of the electromagnetic reconnaissance system. The real and imaginary parts of the matrix are used to construct a 2N-order real symmetric matrix S; and based on the scheduling rule matrix T, the 2N-order real symmetric matrix S and the eigenvector matrix R are iterated M times to obtain the corresponding real symmetric matrix S' and eigenvector matrix R'; finally, based on the elements of matrices S' and R', the N-order Hermitian matrix is ​​obtained. The invention decomposes complex matrices into real matrices and designs a scheduling rule matrix T for iterative calculation, achieving high decomposition accuracy (eigenvalue accuracy can be accurate to 13 decimal places). The same processing method can be applied to eigenvalue decomposition for Hermitian matrices of odd or even dimensions, making it applicable to any processing system that supports parallel computing and thus having a wide range of applications. Attached Figure Description

[0040] Figure 1 This is a flowchart of the Hermitian matrix parallel eigenvalue decomposition method for electromagnetic reconnaissance systems provided in this embodiment of the invention;

[0041] Figure 2 This is a flowchart of the iterative generation of matrices S' and R' provided in an embodiment of the present invention. Detailed Implementation

[0042] The embodiments of the present invention are described in detail below with reference to the accompanying drawings. The embodiments are given for illustrative purposes only and should not be construed as limiting the present invention. The accompanying drawings are for reference and illustration only and do not constitute a limitation on the scope of patent protection of the present invention, because many changes can be made to the present invention without departing from the spirit and scope of the present invention.

[0043] This invention first provides a parallel eigenvalue decomposition method for the Hermitian matrix in electromagnetic reconnaissance systems, such as... Figure 1 The flowchart shown includes the following steps:

[0044] S1. Obtain the N-channel complex signal vectors of the electromagnetic reconnaissance system and construct the N-order Hermitian matrix. ;

[0045] S2. Extract the Nth-order Hermitian matrix. We obtain an N-order real matrix A and an N-order imaginary matrix B from the real and imaginary parts of the matrix; and construct a 2N-order real symmetric matrix S based on matrices A and B.

[0046] S3, Generation The scheduling rule matrix T is such that each row of matrix T consists of 2N elements from 1 to 2N arranged in different orders, and the elements in the first column are all 1;

[0047] S4. Based on the scheduling rule matrix T, perform M iterations on the 2N-order real symmetric matrix S and the eigenvector matrix R to obtain the corresponding real symmetric matrix S' and eigenvector matrix R'.

[0048] S5. Based on the elements of matrices S' and R', obtain the N-order Hermitian matrix. The eigenvalues ​​and the corresponding eigenvectors of each eigenvalue.

[0049] This method constructs a scheduling rule matrix T and performs M iterations on the real symmetric matrix S and the identity matrix R (each iteration has...). The algorithm processes 2N elements in parallel (each step is a parallel process), resulting in highly accurate eigenvalues ​​and eigenvectors. It also boasts high computational speed, short processing time, and strong stability.

[0050] The following provides a more detailed explanation of each step.

[0051] (1) Step S1: Construct an N-order Hermitian matrix

[0052] Taking an 8-channel electromagnetic reconnaissance system as an example, the complex signals from each channel are used to construct a complex signal vector. ,by Obtain the 8th order Hermitian matrix , Columns 1-2, 3-4, 5-6, and 7-8 are shown in Tables 1, 2, 3, and 4, respectively.

[0053] Table 1: Matrix Columns 1-2

[0054]

[0055] Table 2: Matrix Columns 3-4

[0056]

[0057] Table 3: Matrix 5-6 columns

[0058]

[0059] Table 4: Matrix 7-8 columns

[0060]

[0061] The columns 1-4 and columns 5-8 of the N-order real part matrix A corresponding to Tables 1 to 4 are shown in Tables 5 and 6 respectively.

[0062] Table 5: Columns 1-4 of Matrix A

[0063]

[0064] Table 6: Columns 5-8 of Matrix A

[0065]

[0066] (2) Step S2: Based on the N-order Hermitian matrix Construct a 2N-order real symmetric matrix S

[0067] The columns 1-4 and columns 5-8 of the N-order imaginary matrix B corresponding to Tables 1 to 4 are shown in Tables 7 and 8 respectively.

[0068] Table 7: Columns 1-4 of Matrix B

[0069]

[0070] Table 8: Columns 5-8 of Matrix B

[0071]

[0072] In this embodiment, the 2N-order real symmetric matrix S is constructed as follows: .

[0073] The columns 1-4, 5-8, 9-12, and 13-16 of the 16th-order real symmetric matrix S constructed based on Tables 5 to 8 are shown in Tables 9, 10, 11, and 12, respectively.

[0074] Table 9: Columns 1-4 of Matrix S

[0075]

[0076] Table 10: Columns 5-8 of Matrix S

[0077]

[0078] Table 11: Columns 9-12 of Matrix S

[0079]

[0080] Table 12: Columns 13-16 of matrix S

[0081]

[0082] (3) Step S3: Generate The scheduling rule matrix T

[0083] The scheduling rule matrix T has 2N-1 rows and 2N columns, which means that each iteration has 2N-1 steps, and each step processes 2N elements in parallel. These are the row and column indices of the elements processed in the eigenvalue decomposition process, meaning that each row has two elements corresponding to one eigenvalue. Each row of matrix T consists of 2N elements from 1 to 2N arranged in different orders.

[0084] As a preferred implementation, the arrangement rule for rows 1 to N of matrix T is as follows:

[0085] All elements in the first column are 1;

[0086] Starting from the first digit of “2, 4, 6, ..., 2N, 2N-1, 2N-3, 2N-5, ..., 5, 3”, slide the window to the right with a length of N to obtain N sets of data, and fill them sequentially into the first row to the Nth row of the even-numbered column of matrix T.

[0087] Starting from the last character of “3, 5, ..., 2N-1, 2N, 2N-2, ..., 4”, slide the window to the left with a length of N to obtain N-1 sets of data, and fill them sequentially into the even-numbered columns of the N+1th row to the 2N-1th row of matrix T;

[0088] Starting from the last position of “2N-2, 2N-4, ..., 4, 2, 3, 5, ..., 2N-3, 2N-1”, slide the window to the left with a length of N-1 to obtain N-1 sets of data, and fill them sequentially into the remaining odd columns from the first row to the Nth row of matrix T;

[0089] Starting from the first character of “2N-3, 2N-5, ..., 3, 2, 4, ..., 2N-2”, slide the window to the right with a length of N-1 to obtain N-1 sets of data, and fill them sequentially into the remaining odd columns from row N+1 to row 2N-1 of matrix T.

[0090] Taking N=8 as an example, the matrix T generated according to the above rules is shown in Table 13 below.

[0091] Table 13: Matrix T

[0092]

[0093] (4) Step S4: Perform iterative calculation based on the scheduling rule matrix T

[0094] Figure 2 A flowchart illustrating the iterative calculation for step S4. (See attached flowchart.) Figure 2 As shown, the iterative calculation specifically includes the following steps:

[0095] S41. Assign the identity matrix I (2N×2N dimension) to matrix R and initialize the iteration count (clean_num) to 0;

[0096] S42. Determine if the number of iterations (clean_num) is less than M. If so, increment the number of iterations by 1, and initialize the number of iteration steps (step_num) to 0 before proceeding to the next step. Otherwise, output the current matrices S and R as the final real symmetric matrix S' and eigenvector matrix R'.

[0097] S43. Determine if the iteration step number (step_num) is less than 2N-1. If not, return to the previous step. If yes, execute steps S44 to 436.

[0098] S44. Assign the identity matrix I to matrix W;

[0099] S45. Update the element value of the corresponding row and column index in matrix W according to the row and column index pair of the current row of the scheduling rule matrix T and the 2N-order real symmetric matrix S.

[0100] S46. Update matrices S and R based on the updated matrix W, and then return to step S43.

[0101] Step S45 specifically includes the following steps:

[0102] S451. Calculate the row corresponding to the current iteration step based on matrix S and matrix T. i The cosine value of the rotation angle c(j) and the sine value of the rotation angle s(j) ):

[0103] ,

[0104] ,

[0105] ,

[0106] ,

[0107] in, Representation matrix middle OK The element value, Represents a symbolic function. This represents the cotangent value at twice the rotation angle;

[0108] S452, Update matrix W OK List, OK List, OK List, OK The elements of the column are respectively , , , :

[0109]

[0110]

[0111]

[0112] .

[0113] In step S46, the matrix is ​​updated. sum matrix The superscript H indicates the conjugate transpose.

[0114] Following the example in the previous steps, after 10 parallel iterations of the scheduling rule matrix T, the iterated matrices S' and R' are obtained. Columns 1-4, 5-8, 9-12, and 13-16 of matrix S' are shown in Tables 14, 15, 16, and 17, respectively. Columns 1-4, 5-8, 9-12, and 13-16 of matrix R' are shown in Tables 18, 19, 20, and 21, respectively.

[0115] Table 14: Columns 1-4 of matrix S'

[0116]

[0117] Table 15: Columns 5-8 of matrix S'

[0118]

[0119] Table 16: Columns 9-12 of matrix S'

[0120]

[0121] Table 17: Columns 13-16 of matrix S'

[0122]

[0123] Table 18: Columns 1-4 of matrix R'

[0124]

[0125] Table 19: Columns 5-8 of matrix R'

[0126]

[0127] Table 20: Columns 9-12 of matrix R'

[0128]

[0129] Table 21: Columns 13-16 of matrix R'

[0130]

[0131] (5) Step S5: Obtain eigenvalues ​​and eigenvectors

[0132] Step S5 specifically includes the following steps:

[0133] S51. Sort the diagonal elements of the real symmetric matrix S' in ascending order and record the sorted odd index sequence: index(1), index(3), ..., index(2N-1);

[0134] S52. The elements corresponding to the odd indices 1, 3, ..., 2N-1 of the diagonal elements of matrix S' after ascending sorting of its diagonal elements are the matrix itself. The eigenvalues ​​arranged in ascending order;

[0135] S53. Arrange the matrix according to the sorted odd index sequence. The corresponding columns are extracted to form matrix UV. The first N rows of matrix UV are taken to form matrix U, and the last N rows are taken to form matrix V. Then the matrix... Each column is an eigenvector corresponding to the eigenvalues ​​arranged in ascending order.

[0136] Continuing the previous example, when N=8, the diagonal elements of the matrix S' obtained in step S51 are sorted in ascending order as index(1), index(3), ..., index(15). The elements corresponding to the odd indices 1, 3, ..., 15 are the matrix elements. The eigenvalues ​​are arranged in ascending order, as shown in Table 22.

[0137] Table 22: Eigenvalues

[0138]

[0139] Extract the corresponding columns from matrix R' to form matrix UV. Take the first 8 rows of matrix UV to form matrix U, and the last 8 rows to form matrix V, thus obtaining matrix U. ,matrix Each column of the matrix represents the eigenvector corresponding to the eigenvalues ​​arranged in ascending order. Columns 1-2, 3-4, 5-6, and 7-8 are shown in Tables 23, 24, 25, and 26, respectively.

[0140] Table 23: Matrix Columns 1-2

[0141]

[0142] Table 24: Matrix Columns 3-4

[0143]

[0144] Table 25: Matrix 5-6 columns

[0145]

[0146] Table 26: Matrix 7-8 columns

[0147]

[0148] Use the Matlab function eig() to process the matrix The eigenvalues ​​are decomposed and arranged in ascending order as shown in Table 27. Based on Tables 22 and 27, the absolute errors of the eigenvalues ​​can be calculated as shown in Table 28. Table 28 shows that using this method for parallel eigenvalue decomposition of the Hermitian matrix achieves eigenvalue accuracy to 13 decimal places.

[0149] Table 27: Ascending sort characteristic values ​​obtained using Matlab functions

[0150]

[0151] Table 28: Absolute Error of Eigenvalues

[0152]

[0153] It should be noted that the various processes shown above can be used to reorder, add, or delete steps. For example, the steps described in this invention can be executed in parallel, sequentially, or in different orders, as long as the desired result of the technical solution of this invention can be achieved. This embodiment does not impose any limitations on these steps.

[0154] The embodiments described in this invention can be implemented in computing systems that include backend components (e.g., as a data server), or computing systems that include middleware components (e.g., an application server), or computing systems that include frontend components (e.g., a user computer with a graphical user interface or web browser through which a user can interact with the embodiments of the systems and technologies described herein), or any combination of such backend, middleware, or frontend components. The components of the system can be interconnected via digital data communication of any form or medium (e.g., a communication network). Examples of communication networks include local area networks (LANs), wide area networks (WANs), blockchain networks, and the Internet.

[0155] Computer programs for implementing the methods of the present invention may be written in any combination of one or more programming languages ​​and stored in a computer-readable storage medium. These computer programs may be provided to a processor of a general-purpose computer, a special-purpose computer, or other programmable data processing apparatus, such that when executed by the processor, the computer programs cause the functions / operations specified in the flowcharts and / or block diagrams to be performed. The computer programs may be executed entirely on a machine, partially on a machine, or as a standalone software package, partially on a machine and partially on a remote machine, or entirely on a remote machine or server.

[0156] Computer-readable storage media can be tangible media that may contain or store computer programs for use by or in conjunction with an instruction execution system, apparatus, or device. Computer-readable storage media can be, but is not limited to, electronic, magnetic, optical, electromagnetic, infrared, or semiconductor systems, apparatus, or devices, or any suitable combination of the foregoing. Alternatively, computer-readable storage media can be machine-readable signal media. More specific examples of machine-readable storage media include electrical connections based on one or more wires, portable computer disks, hard disks, random access memory (RAM), read-only memory (ROM), erasable programmable read-only memory (EPROM or flash memory), optical fibers, optical disc read-only memory (CD-ROM), optical storage devices, magnetic storage devices, or any suitable combination of the foregoing.

[0157] It should also be noted that this method is applicable to other fields of mathematics, physics, and engineering.

[0158] In summary, the parallel eigenvalue decomposition method for the Hermitian matrix of an electromagnetic reconnaissance system provided in this embodiment of the invention is based on the N-order Hermitian matrix of the electromagnetic reconnaissance system. The real and imaginary parts of the matrix are used to construct a 2N-order real symmetric matrix S; and based on the scheduling rule matrix T, the 2N-order real symmetric matrix S and the eigenvector matrix R are iterated M times to obtain the corresponding real symmetric matrix S' and eigenvector matrix R'; finally, based on the elements of matrices S' and R', the N-order Hermitian matrix is ​​obtained. The invention decomposes complex matrices into real matrices and designs a scheduling rule matrix T for iterative calculation, achieving high decomposition accuracy (eigenvalue accuracy can be accurate to 13 decimal places). The same processing method can be applied to eigenvalue decomposition for Hermitian matrices of odd or even dimensions, making it applicable to any processing system that supports parallel computing and thus having a wide range of applications.

[0159] The above embodiments are preferred embodiments of the present invention, but the embodiments of the present invention are not limited to the above embodiments. Any changes, modifications, substitutions, combinations, or simplifications made without departing from the spirit and principle of the present invention shall be considered equivalent substitutions and shall be included within the protection scope of the present invention.

Claims

1. A parallel eigenvalue decomposition method for the Hermitian matrix in electromagnetic reconnaissance systems, characterized in that, include: S1. Obtain the N-channel complex signal vectors of the electromagnetic reconnaissance system and construct the N-order Hermitian matrix. ; S2. Extract the Nth-order Hermitian matrix. We obtain an N-order real matrix A and an N-order imaginary matrix B from the real and imaginary parts of the matrix; and construct a 2N-order real symmetric matrix S based on matrices A and B. S3, Generation The scheduling rule matrix T is such that each row of matrix T consists of 2N elements from 1 to 2N arranged in different orders, and the elements in the first column are all 1; S4. Based on the scheduling rule matrix T, perform M iterations on the 2N-order real symmetric matrix S and the eigenvector matrix R to obtain the corresponding real symmetric matrix S' and eigenvector matrix R'. Step S4 specifically includes the following steps: S41. Assign the 2N×2N identity matrix I to matrix R and initialize the iteration count to 0; S42. Determine if the number of iterations is less than M. If so, increment the number of iterations by 1, initialize the number of iterations to 0, and proceed to the next step. Otherwise, output the current matrices S and R as the final real symmetric matrix S' and eigenvector matrix R'. S43. Determine if the number of iterations is less than 2N-1. If not, return to the previous step. If yes, execute steps S44 to S46. S44. Assign the identity matrix I to matrix W; S45. Update the element value of the corresponding row and column index in matrix W according to the row and column index pair of the current row of the scheduling rule matrix T and the 2N-order real symmetric matrix S. S46. Update matrices S and R based on the updated matrix W, and then return to step S43; Step S45 specifically includes the following steps: S451. Calculate the cosine value c(j) and sine value s(j) of the rotation angle corresponding to the current iteration step based on matrix S and matrix T. ; S452, based on , , Update W OK List, OK List, OK List, OK Column elements; In step S451, , Calculated by the following formula: ， ， ， ， in, This represents the cotangent value at twice the rotation angle. Representation matrix middle OK The element value, Represents a symbolic function; In step S452, the matrix W's OK List, OK List, OK List, OK The elements of the column are updated respectively. , , , ; In step S46, the matrix is ​​updated. sum matrix The superscript H indicates conjugate transpose; S5. Based on the elements of matrices S' and R', obtain the N-order Hermitian matrix. The eigenvalues ​​and the corresponding eigenvectors of each eigenvalue.

2. The Hermitian matrix parallel eigenvalue decomposition method for electromagnetic reconnaissance systems according to claim 1, characterized in that, In step S2, matrix S is constructed as follows: .

3. The Hermitian matrix parallel eigenvalue decomposition method for electromagnetic reconnaissance systems according to claim 1, characterized in that, In step S3, the arrangement rule for rows 1 to N of matrix T is as follows: All elements in the first column are 1; Starting from the first character of "2, 4, 6, ..., 2N, 2N-1, 2N-3, 2N-5, ..., 5, 3", slide the window to the right with a length of N to obtain N sets of data, and fill them sequentially into the first row to the Nth row of the even-numbered column of matrix T. Starting from the last character of "3, 5, ..., 2N-1, 2N, 2N-2, ..., 4", slide the window to the left with a length of N to obtain N-1 sets of data, and fill them sequentially into the even-numbered columns of the N+1th row to the 2N-1th row of matrix T; Starting from the last position of "2N-2, 2N-4, ..., 4, 2, 3, 5, ..., 2N-3, 2N-1", slide the window to the left with a length of N-1 to obtain N-1 sets of data, and fill them sequentially into the remaining odd columns from the first row to the Nth row of matrix T; Starting from the first character of "2N-3, 2N-5, ..., 3, 2, 4, ..., 2N-2", slide the window to the right with a length of N-1 to obtain N-1 sets of data, and fill them sequentially into the remaining odd columns from row N+1 to row 2N-1 of matrix T.

4. The parallel eigenvalue decomposition method for the Hermitian matrix in an electromagnetic reconnaissance system according to claim 1, characterized in that, Step S5 specifically includes the following steps: S51. Sort the diagonal elements of the real symmetric matrix S' in ascending order and record the sorted odd index sequence: index(1), index(3), ..., index(2N-1); S52. The elements corresponding to the odd indices 1, 3, ..., 2N-1 of the diagonal elements of matrix S' after ascending sorting of its diagonal elements are the matrix itself. The eigenvalues ​​arranged in ascending order; S53. Arrange the matrix according to the sorted odd index sequence. The corresponding columns are extracted to form matrix UV. The first N rows of matrix UV are taken to form matrix U, and the last N rows are taken to form matrix V. Then the matrix... Each column is an eigenvector corresponding to the eigenvalues ​​arranged in ascending order.

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