Singular Value Decomposition Calculator
Results:
Here’s a comprehensive table summarizing the key concepts and components of SVD:
| Concept | Description |
|---|---|
| Definition | Decomposition of a matrix AAA into three matrices UUU, Σ\SigmaΣ, and VTV^TVT such that A=UΣVTA = U \Sigma V^TA=UΣVT. |
| Matrix AAA | The original matrix of dimensions m×nm \times nm×n. |
| Matrix UUU | An m×mm \times mm×m orthogonal matrix, whose columns are the left singular vectors of AAA. |
| Matrix Σ\SigmaΣ | An m×nm \times nm×n diagonal matrix, with non-negative real numbers on the diagonal (the singular values of AAA). |
| Matrix VTV^TVT | An n×nn \times nn×n orthogonal matrix, whose columns are the right singular vectors of AAA. |
| Singular Values | The diagonal elements of Σ\SigmaΣ, representing the “strength” or “magnitude” of each corresponding dimension. |
| Orthogonality | UUU and VVV are orthogonal matrices, meaning UTU=IU^T U = IUTU=I and VTV=IV^T V = IVTV=I, where III is the identity matrix. |
| Rank | The number of non-zero singular values in Σ\SigmaΣ, equal to the rank of the matrix AAA. |
| Applications | Principal Component Analysis (PCA), image compression, noise reduction, and solving linear systems, among others. |