CA2588176A1 - Eigenvalue decomposition and singular value decomposition of matrices using jacobi rotation - Google Patents
Eigenvalue decomposition and singular value decomposition of matrices using jacobi rotation Download PDFInfo
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Abstract
Techniques for decomposing matrices using Jacobi rotation are described.
Multiple iterations of Jacobi rotation are performed on a first matrix of complex values with multiple Jacobi rotation matrices of complex values to zero out the off-diagonal elements in the first matrix. For each iteration, a submatrix may be formed based on the first matrix and decomposed to obtain eigenvectors for the submatrix, and a Jacobi rotation matrix may be formed with the eigenvectors and used to update the first matrix. A second matrix of complex values, which contains orthogonal vectors, is derived based on the Jacobi rotation matrices. For eigenvalue decomposition, a third matrix of eigenvalues may be derived based on the Jacobi rotation matrices. For singular value decomposition, a fourth matrix with left singular vectors and a matrix of singular values may be derived based on the Jacobi rotation matrices.
Multiple iterations of Jacobi rotation are performed on a first matrix of complex values with multiple Jacobi rotation matrices of complex values to zero out the off-diagonal elements in the first matrix. For each iteration, a submatrix may be formed based on the first matrix and decomposed to obtain eigenvectors for the submatrix, and a Jacobi rotation matrix may be formed with the eigenvectors and used to update the first matrix. A second matrix of complex values, which contains orthogonal vectors, is derived based on the Jacobi rotation matrices. For eigenvalue decomposition, a third matrix of eigenvalues may be derived based on the Jacobi rotation matrices. For singular value decomposition, a fourth matrix with left singular vectors and a matrix of singular values may be derived based on the Jacobi rotation matrices.
Claims (45)
1. An apparatus comprising:
at least one processor configured to perform a plurality of iterations of Jacobi rotation on a first matrix of complex values with a plurality of Jacobi rotation matrices of complex values, and to derive a second matrix of complex values based on the plurality of Jacobi rotation matrices, the second matrix comprising orthogonal vectors; and a memory coupled to the at least one processor.
at least one processor configured to perform a plurality of iterations of Jacobi rotation on a first matrix of complex values with a plurality of Jacobi rotation matrices of complex values, and to derive a second matrix of complex values based on the plurality of Jacobi rotation matrices, the second matrix comprising orthogonal vectors; and a memory coupled to the at least one processor.
2. ~The apparatus of claim 1, wherein for each of the plurality of iterations the at least one processor is configured to form a submatrix based on the first matrix, to decompose the submatrix to obtain eigenvectors for the submatrix, to form a Jacobi rotation matrix with the eigenvectors, and to update the first matrix with the Jacobi rotation matrix.
3. ~The apparatus of claim 2, wherein for each of the plurality of iterations, the at least one processor is configured to order the eigenvectors for the submatrix based on eigenvalues for the submatrix.
4. ~The apparatus of claim 1, wherein the at least one processor is configured to derive a third matrix of eigenvalues based on the plurality of Jacobi rotation matrices.
5. ~The apparatus of claim 1, wherein the at least one processor is configured to derive a third matrix of complex values based on the plurality of Jacobi rotation matrices; and to derive a fourth matrix with orthogonal vectors based on the third matrix.
6. ~The apparatus of claim 5, wherein the at least one processor is configured to derive a matrix of singular values based on the third matrix.
7. ~The apparatus of claim 1, wherein the at least one processor is configured to derive a third matrix with orthogonal vectors based on the plurality of Jacobi rotation matrices.
8. ~The apparatus of claim 7, wherein the at least one processor is configured to derive a matrix of singular values based on the plurality of Jacobi rotation matrices.
9. ~The apparatus of claim 1, wherein the at least one processor is configured to select different values for row and column indices of the first matrix for the plurality of iterations of the Jacobi rotation.
10. ~The apparatus of claim 1, wherein for each of the plurality of iterations the at least one processor is configured to identify a largest off-diagonal element in the first matrix, and to perform the Jacobi rotation based on the largest off-diagonal element.
11. ~The apparatus of claim 1, wherein the at least one processor is configured to terminate the Jacobi rotation on the first matrix after a predetermined number of iterations.
12. ~The apparatus of claim 1, wherein the at least one processor is configured to determine whether an error criterion is satisfied, and to terminate the plurality of iterations of the Jacobi rotation upon satisfaction of the error criterion.
13. ~The apparatus of claim 1, wherein the first matrix has a dimension larger than 2 × 2.
14. ~A method comprising:
performing a plurality of iterations of Jacobi rotation on a first matrix of complex values with a plurality of Jacobi rotation matrices of complex values;
and deriving a second matrix of complex values based on the plurality of Jacobi rotation matrices, the second matrix comprising orthogonal vectors.
performing a plurality of iterations of Jacobi rotation on a first matrix of complex values with a plurality of Jacobi rotation matrices of complex values;
and deriving a second matrix of complex values based on the plurality of Jacobi rotation matrices, the second matrix comprising orthogonal vectors.
15. ~The method of claim 14, wherein the performing the plurality of iterations of Jacobi rotation on the first matrix comprises, for each iteration, forming a submatrix based on the first matrix, decomposing the submatrix to obtain eigenvectors for the submatrix, forming a Jacobi rotation matrix with the eigenvectors, and updating the first matrix with the Jacobi rotation matrix.
16. ~The method of claim 14, further comprising:
deriving a third matrix of complex values based on the plurality of Jacobi rotation matrices; and deriving a fourth matrix with orthogonal vectors based on the third matrix.
deriving a third matrix of complex values based on the plurality of Jacobi rotation matrices; and deriving a fourth matrix with orthogonal vectors based on the third matrix.
17. ~The method of claim 14, further comprising:
deriving a third matrix with orthogonal vectors based on the plurality of Jacobi rotation matrices.
deriving a third matrix with orthogonal vectors based on the plurality of Jacobi rotation matrices.
18. ~An apparatus comprising:
means for performing a plurality of iterations of Jacobi rotation on a first matrix of complex values with a plurality of Jacobi rotation matrices of complex values; and means for deriving a second matrix of complex values based on the plurality of Jacobi rotation matrices, the second matrix comprising orthogonal vectors.
means for performing a plurality of iterations of Jacobi rotation on a first matrix of complex values with a plurality of Jacobi rotation matrices of complex values; and means for deriving a second matrix of complex values based on the plurality of Jacobi rotation matrices, the second matrix comprising orthogonal vectors.
19. ~The apparatus of claim 18, wherein the means for performing the plurality of iterations of Jacobi rotation on the first matrix comprises, for each iteration, means for forming a submatrix based on the first matrix, means for decomposing the submatrix to obtain eigenvectors for the submatrix, means for forming a Jacobi rotation matrix with the eigenvectors, and means for updating the first matrix with the Jacobi rotation matrix.
20. ~The apparatus of claim 18, further comprising:
means for deriving a third matrix of complex values based on the plurality of Jacobi rotation matrices; and means for deriving a fourth matrix with orthogonal vectors based on the third matrix.
means for deriving a third matrix of complex values based on the plurality of Jacobi rotation matrices; and means for deriving a fourth matrix with orthogonal vectors based on the third matrix.
21. The apparatus of claim 18, further comprising:
means for deriving a third matrix with orthogonal vectors based on the plurality of Jacobi rotation matrices.
means for deriving a third matrix with orthogonal vectors based on the plurality of Jacobi rotation matrices.
22. An apparatus comprising:
at least one processor configured to initialize a first matrix to an identity matrix, to initialize a second matrix to a Hermitian matrix of complex values, to perform a plurality of iterations of Jacobi rotation on the second matrix by forming a Jacobi rotation matrix of complex values for each iteration based on the second matrix, and updating the first and second matrices for each iteration based on the Jacobi rotation matrix for the iteration, to provide the first matrix as a matrix of eigenvectors, and to provide the second matrix as a matrix of eigenvalues; and a memory coupled to the at least one processor.
at least one processor configured to initialize a first matrix to an identity matrix, to initialize a second matrix to a Hermitian matrix of complex values, to perform a plurality of iterations of Jacobi rotation on the second matrix by forming a Jacobi rotation matrix of complex values for each iteration based on the second matrix, and updating the first and second matrices for each iteration based on the Jacobi rotation matrix for the iteration, to provide the first matrix as a matrix of eigenvectors, and to provide the second matrix as a matrix of eigenvalues; and a memory coupled to the at least one processor.
23. The apparatus of claim 22, wherein for each of the plurality of iterations the at least one processor is configured to form a submatrix based on the second matrix, to decompose the submatrix to obtain eigenvectors for the submatrix, and to form the Jacobi rotation matrix with the eigenvectors for the submatrix.
24. An apparatus comprising:
means for initializing a first matrix to an identity matrix;
means for initializing a second matrix to a Hermitian matrix of complex values;
means for performing a plurality of iterations of Jacobi rotation on the second matrix, comprising means for forming a Jacobi rotation matrix of complex values for each iteration based on the second matrix, and means for updating the first and second matrices for each iteration based on the Jacobi rotation matrix for the iteration;
means for providing the first matrix as a matrix of eigenvectors; and means for providing the second matrix as a matrix of eigenvalues.
means for initializing a first matrix to an identity matrix;
means for initializing a second matrix to a Hermitian matrix of complex values;
means for performing a plurality of iterations of Jacobi rotation on the second matrix, comprising means for forming a Jacobi rotation matrix of complex values for each iteration based on the second matrix, and means for updating the first and second matrices for each iteration based on the Jacobi rotation matrix for the iteration;
means for providing the first matrix as a matrix of eigenvectors; and means for providing the second matrix as a matrix of eigenvalues.
25. The apparatus of claim 24, wherein the means for forming the Jacobi rotation matrix of complex values for each iteration comprises means for forming a submatrix based on the second matrix, means for decomposing the submatrix to obtain eigenvectors for the submatrix, and means for forming the Jacobi rotation matrix with the eigenvectors for the submatrix.
26. An apparatus comprising:
at least one processor is configured to initialize a first matrix to an identity matrix, to initialize a second matrix to a matrix of complex values, to perform a plurality of iterations of Jacobi rotation on the second matrix by forming a Jacobi rotation matrix for each iteration based on the second matrix, and updating the first and second matrices for each iteration based on the Jacobi rotation matrix for the iteration, and to provide the first matrix as a matrix of right singular vectors; and a memory coupled to the at least one processor.
at least one processor is configured to initialize a first matrix to an identity matrix, to initialize a second matrix to a matrix of complex values, to perform a plurality of iterations of Jacobi rotation on the second matrix by forming a Jacobi rotation matrix for each iteration based on the second matrix, and updating the first and second matrices for each iteration based on the Jacobi rotation matrix for the iteration, and to provide the first matrix as a matrix of right singular vectors; and a memory coupled to the at least one processor.
27. The apparatus of claim 26, wherein for each of the plurality of iterations the at least one processor is configured to form a submatrix based on the second matrix, to decompose the submatrix to obtain eigenvectors for the submatrix, and to form the Jacobi rotation matrix with the eigenvectors.
28. The apparatus of claim 26, wherein the at least one processor is configured to derive a matrix of singular values based on the second matrix.
29. The apparatus of claim 26, wherein the at least one processor is configured to derive a matrix of left singular vectors based on second matrix.
30. An apparatus comprising:
means for initializing a first matrix to an identity matrix;
means for initializing a second matrix to a matrix of complex values;
means for performing a plurality of iterations of Jacobi rotation on the second matrix, comprising means for forming a Jacobi rotation matrix for each iteration based on the second matrix, and means for updating the first and second matrices for each iteration based on the Jacobi rotation matrix for the iteration; and means for provide the first matrix as a matrix of right singular vectors.
means for initializing a first matrix to an identity matrix;
means for initializing a second matrix to a matrix of complex values;
means for performing a plurality of iterations of Jacobi rotation on the second matrix, comprising means for forming a Jacobi rotation matrix for each iteration based on the second matrix, and means for updating the first and second matrices for each iteration based on the Jacobi rotation matrix for the iteration; and means for provide the first matrix as a matrix of right singular vectors.
31. The apparatus of claim 30, wherein the means for forming the Jacobi rotation matrix for each iteration comprises means for forming a submatrix based on the second matrix, means for decomposing the submatrix to obtain eigenvectors for the submatrix, and means for forming the Jacobi rotation matrix with the eigenvectors.
32. An apparatus comprising:
at least one processor is configured to initialize a first matrix to an identity matrix, to initialize a second matrix to the identity matrix, to initialize a third matrix to a matrix of complex values, to perform a plurality of iterations of Jacobi rotation on the third matrix by, for each iteration, forming a first Jacobi rotation matrix based on the third matrix, forming a second Jacobi rotation matrix based on the third matrix, updating the first matrix based on the first Jacobi rotation matrix, updating the second matrix based on the second Jacobi rotation matrix, and updating the third matrix based on the first and second Jacobi rotation matrices, and to provide the second matrix as a matrix of left singular vectors; and a memory coupled to the at least one processor.
at least one processor is configured to initialize a first matrix to an identity matrix, to initialize a second matrix to the identity matrix, to initialize a third matrix to a matrix of complex values, to perform a plurality of iterations of Jacobi rotation on the third matrix by, for each iteration, forming a first Jacobi rotation matrix based on the third matrix, forming a second Jacobi rotation matrix based on the third matrix, updating the first matrix based on the first Jacobi rotation matrix, updating the second matrix based on the second Jacobi rotation matrix, and updating the third matrix based on the first and second Jacobi rotation matrices, and to provide the second matrix as a matrix of left singular vectors; and a memory coupled to the at least one processor.
33. The apparatus of claim 32, wherein for each of the plurality of iterations the at least one processor is configured to form a first submatrix based on the third matrix, to decompose the first submatrix to obtain eigenvectors for the first submatrix, and to form the first Jacobi rotation matrix with the eigenvectors for the first submatrix.
34. The apparatus of claim 33, wherein for each of the plurality of iterations the at least one processor is configured to form a second submatrix based on the third matrix, to decompose the second submatrix to obtain eigenvectors for the second submatrix, and to form the second Jacobi rotation matrix with the eigenvectors for the second submatrix.
35. The apparatus of claim 32, wherein the at least one processor is configured to derive a matrix of right singular vectors based on the first matrix.
36. The apparatus of claim 32, wherein the at least one processor is configured to derive a matrix of singular values based on the third matrix.
37. An apparatus comprising:
means for initializing a first matrix to an identity matrix;
means for initializing a second matrix to the identity matrix;
means for initializing a third matrix to a matrix of complex values, means for performing a plurality of iterations of Jacobi rotation on the third matrix comprising, for each iteration, means for forming a first Jacobi rotation matrix based on the third matrix, means for forming a second Jacobi rotation matrix based on the third matrix, means for updating the first matrix based on the first Jacobi rotation matrix, means for updating the second matrix based on the second Jacobi rotation matrix, and means for updating the third matrix based on the first and second Jacobi rotation matrices; and means for providing the second matrix as a matrix of left singular vectors.
means for initializing a first matrix to an identity matrix;
means for initializing a second matrix to the identity matrix;
means for initializing a third matrix to a matrix of complex values, means for performing a plurality of iterations of Jacobi rotation on the third matrix comprising, for each iteration, means for forming a first Jacobi rotation matrix based on the third matrix, means for forming a second Jacobi rotation matrix based on the third matrix, means for updating the first matrix based on the first Jacobi rotation matrix, means for updating the second matrix based on the second Jacobi rotation matrix, and means for updating the third matrix based on the first and second Jacobi rotation matrices; and means for providing the second matrix as a matrix of left singular vectors.
38. The apparatus of claim 37, wherein the means for forming the first Jacobi rotation matrix comprises means for forming a first submatrix based on the third matrix, means for decomposing the first submatrix to obtain eigenvectors for the first submatrix, and means for forming the first Jacobi rotation matrix with the eigenvectors for the first submatrix.
39. The apparatus of claim 38, wherein the means for forming the second Jacobi rotation matrix comprises means for forming a second submatrix based on the third matrix, means for decomposing the second submatrix to obtain eigenvectors for the second submatrix, and means for forming the second Jacobi rotation matrix with the eigenvectors for the second submatrix.
40. An apparatus comprising:
at least one processor is configured to perform a first plurality of iterations of Jacobi rotation on a first matrix of complex values to obtain a first unitary matrix with orthogonal vectors, and to perform a second plurality of iterations of the Jacobi rotation on a second matrix of complex values to obtain a second unitary matrix with orthogonal vectors, wherein the first unitary matrix is used as an initial solution for the second unitary matrix; and a memory coupled to the at least one processor.
at least one processor is configured to perform a first plurality of iterations of Jacobi rotation on a first matrix of complex values to obtain a first unitary matrix with orthogonal vectors, and to perform a second plurality of iterations of the Jacobi rotation on a second matrix of complex values to obtain a second unitary matrix with orthogonal vectors, wherein the first unitary matrix is used as an initial solution for the second unitary matrix; and a memory coupled to the at least one processor.
41. The apparatus of claim 40, wherein the at least one processor is configured to perform a third plurality of iterations of the Jacobi rotation on a third matrix of complex values to obtain a third unitary matrix with orthogonal vectors, wherein the second unitary matrix is used as an initial solution for the third unitary matrix.
42. The apparatus of claim 40, wherein the first and second matrices of complex values are channel response matrices for two frequency subbands.
43. The apparatus of claim 40, wherein the first and second matrices of complex values are channel response matrices for two time intervals.
44. An apparatus comprising:
means for performing a first plurality of iterations of Jacobi rotation on a first matrix of complex values to obtain a first unitary matrix with orthogonal vectors; and means for performing a second plurality of iterations of the Jacobi rotation on a second matrix of complex values to obtain a second unitary matrix with orthogonal vectors, wherein the first unitary matrix is used as an initial solution for the second unitary matrix.
means for performing a first plurality of iterations of Jacobi rotation on a first matrix of complex values to obtain a first unitary matrix with orthogonal vectors; and means for performing a second plurality of iterations of the Jacobi rotation on a second matrix of complex values to obtain a second unitary matrix with orthogonal vectors, wherein the first unitary matrix is used as an initial solution for the second unitary matrix.
45. The apparatus of claim 44, wherein the first and second matrices of complex values are channel response matrices for two frequency subbands.
Applications Claiming Priority (3)
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| US62832404P | 2004-11-15 | 2004-11-15 | |
| US60/628,324 | 2004-11-15 | ||
| PCT/US2005/041783 WO2006053340A2 (en) | 2004-11-15 | 2005-11-15 | Eigenvalue decomposition and singular value decomposition of matrices using jacobi rotation |
Publications (2)
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| CA2588176A1 true CA2588176A1 (en) | 2006-05-18 |
| CA2588176C CA2588176C (en) | 2012-10-16 |
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| IN (1) | IN2012DN01928A (en) |
| TW (1) | TWI407320B (en) |
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| TWI407320B (en) | 2013-09-01 |
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| AR051497A1 (en) | 2007-01-17 |
| KR101084792B1 (en) | 2011-11-21 |
| CN101438277A (en) | 2009-05-20 |
| CN101390351A (en) | 2009-03-18 |
| KR20090115822A (en) | 2009-11-06 |
| CA2588176C (en) | 2012-10-16 |
| JP2008521294A (en) | 2008-06-19 |
| JP4648401B2 (en) | 2011-03-09 |
| KR20070086178A (en) | 2007-08-27 |
| WO2006053340A3 (en) | 2008-07-31 |
| TW200703039A (en) | 2007-01-16 |
| CN101390351B (en) | 2012-10-10 |
| IN2012DN01928A (en) | 2015-07-24 |
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