A Matrix-Valued Inner Product for Matrix-Valued Signals and Matrix-Valued Lattices
Fuente:
arXiv
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| Formato: | Preprint |
| Publicado: |
2020
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| _version_ | 1866912577054834688 |
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| author | Xia, Xiang-Gen |
| author_facet | Xia, Xiang-Gen |
| contents | A matrix-valued inner product was proposed before to construct orthonormal matrix-valued wavelets for matrix-valued signals. It introduces a weaker orthogonality for matrix-valued signals than the orthogonality of all components in a matrix that is commonly used in orthogonal multiwavelet constructions. With the weaker orthogonality, it is easier to construct orthonormal matrix-valued wavelets. In this paper, we re-study the matrix-valued inner product more from the inner product viewpoint that is more fundamental and propose a new but equivalent norm for matrix-valued signals. We show that although it is not scalar-valued, it maintains most of the scalar-valued inner product properties. We introduce a new linear independence concept for matrix-valued signals and present some related properties. We then present the Gram-Schmidt orthonormalization procedure for a set of linearly independent matrix-valued signals. Finally we define matrix-valued lattices. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2010_03075 |
| institution | arXiv |
| publishDate | 2020 |
| record_format | arxiv |
| spellingShingle | A Matrix-Valued Inner Product for Matrix-Valued Signals and Matrix-Valued Lattices Xia, Xiang-Gen Functional Analysis Signal Processing A matrix-valued inner product was proposed before to construct orthonormal matrix-valued wavelets for matrix-valued signals. It introduces a weaker orthogonality for matrix-valued signals than the orthogonality of all components in a matrix that is commonly used in orthogonal multiwavelet constructions. With the weaker orthogonality, it is easier to construct orthonormal matrix-valued wavelets. In this paper, we re-study the matrix-valued inner product more from the inner product viewpoint that is more fundamental and propose a new but equivalent norm for matrix-valued signals. We show that although it is not scalar-valued, it maintains most of the scalar-valued inner product properties. We introduce a new linear independence concept for matrix-valued signals and present some related properties. We then present the Gram-Schmidt orthonormalization procedure for a set of linearly independent matrix-valued signals. Finally we define matrix-valued lattices. |
| title | A Matrix-Valued Inner Product for Matrix-Valued Signals and Matrix-Valued Lattices |
| topic | Functional Analysis Signal Processing |
| url | https://arxiv.org/abs/2010.03075 |