"Gram matrix"의 두 판 사이의 차이
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Theorem 7.2.10 Let | Theorem 7.2.10 Let | ||
\(\displaystyle v_{1},\ldots ,v_{m}\) be vectors in an inner product space V with inner product \(\displaystyle \langle {\cdot ,\cdot }\rangle \) and let \(\displaystyle G=[\langle {v_{j},v_{i}}\rangle ]_{i,j=1}^{m}\in M_{m}\). Then <br /> | \(\displaystyle v_{1},\ldots ,v_{m}\) be vectors in an inner product space V with inner product \(\displaystyle \langle {\cdot ,\cdot }\rangle \) and let \(\displaystyle G=[\langle {v_{j},v_{i}}\rangle ]_{i,j=1}^{m}\in M_{m}\). Then <br /> | ||
| − | (a) G is [[Hermitian Matrix|Hermitian]] and positive-semidefinite<br /> | + | (a) G is [[Hermitian Matrix|Hermitian]] and [[positive definite|positive-semidefinite]]<br /> |
| − | (b) G is positive-definite if and only if the vectors \(\displaystyle v_{1},\ldots ,v_{m}\) are linearly-independent.<br /> | + | (b) G is [[positive definite|positive-definite]] if and only if the vectors \(\displaystyle v_{1},\ldots ,v_{m}\) are linearly-independent.<br /> |
(c) \(\displaystyle \operatorname {rank} G=\dim \operatorname {span} \{v_{1},\ldots ,v_{m}\}\) | (c) \(\displaystyle \operatorname {rank} G=\dim \operatorname {span} \{v_{1},\ldots ,v_{m}\}\) | ||
2017년 7월 10일 (월) 01:58 판
Gram matrix \( G = V^T V\) where \(V\) is a matrix.(for finite-dimensional vectors with the usual Euclidean dot product) [1]
Horn & Johnson 2013[1], p. 441[2]
Theorem 7.2.10 Let
\(\displaystyle v_{1},\ldots ,v_{m}\) be vectors in an inner product space V with inner product \(\displaystyle \langle {\cdot ,\cdot }\rangle \) and let \(\displaystyle G=[\langle {v_{j},v_{i}}\rangle ]_{i,j=1}^{m}\in M_{m}\). Then
(a) G is Hermitian and positive-semidefinite
(b) G is positive-definite if and only if the vectors \(\displaystyle v_{1},\ldots ,v_{m}\) are linearly-independent.
(c) \(\displaystyle \operatorname {rank} G=\dim \operatorname {span} \{v_{1},\ldots ,v_{m}\}\)
- ↑ Horn, Roger A.; Johnson, Charles R. (2013). "7.2 Characterizations and Properties". Matrix Analysis (Second Edition). Cambridge University Press. ISBN 978-0-521-83940-2.
- ↑ https://en.wikipedia.org/wiki/Gramian_matrix#cite_note-1