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词汇 Singular value decomposition
分类 英语词汇 英语翻译词典
释义

Singular value decomposition

中文百科

奇异值分解

奇异值分解（singular value decomposition）是线性代数中一种重要的矩阵分解，在信号处理、统计学等领域有重要应用。奇异值分解在某些方面与对称矩阵或厄米矩阵基于特征矢量的对角化类似。然而这两种矩阵分解尽管有其相关性，但还是有明显的不同。对称阵特征矢量分解的基础是谱分析，而奇异值分解则是谱分析理论在任意矩阵上的推广。

英语百科

Singular value decomposition 奇异值分解

Visualization of the SVD of a two-dimensional, real shearing matrix M. First, we see the unit disc in blue together with the two canonical unit vectors. We then see the action of M, which distorts the disk to an ellipse. The SVD decomposes M into three simple transformations: an initial rotation V∗, a scaling Σ along the coordinate axes, and a final rotation U. The lengths σ1 and σ2 of the semi-axes of the ellipse are the singular values of M, namely Σ1,1 and Σ2,2.
The image shows: Upper Left: The unit disc with the two canonical unit vectors Upper Right: Unit disc transformed with M and singular Values σ1 and σ2 indicated Lower Left: The action of V∗ on the unit disc. This is just a rotation. Lower Right: The action of ΣV∗ on the unit disc. Sigma scales in vertically and horizontally. In this special case, the singular values are Phi and 1/Phi where Phi is the Golden ratio. V∗ is a (counter clockwise) rotation by an angle alpha where alpha satisfies tan(alpha) = -Phi. U is a rotation by an angle beta with tan(beta) = Phi-1

In linear algebra, the singular value decomposition (SVD) is a factorization of a real or complex matrix. It is the generalization of the eigendecomposition of a positive semidefinite normal matrix (for example, a symmetric matrix with positive eigenvalues) to any m \times n matrix via an extension of polar decomposition. It has many useful applications in signal processing and statistics.
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