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SVD is used in signal processing, data reduction, solving inverse problems, and principal component analysis among other applications.

In image compression, SVD can reduce the amount of information needed to reconstruct an image by keeping only a few of the largest singular values and corresponding singular vectors, significantly reducing the data size without substantial quality loss.

The diagonal matrix $\Sigma$ in SVD contains the singular values, which are the lengths of the semi-axes of the ellipsoid represented by the matrix $A$, ordered from largest to smallest.

Full SVD includes all the singular values (including zeros) and corresponding vectors, resulting in $U$ and $V$ matrices of size equal to the original matrix. In contrast, reduced SVD omits the zero singular values and the corresponding vectors.

The pseudoinverse of a matrix $A$, denoted $A^+$, can be computed using SVD by taking $A^+ = V\Sigma^+U^*$ where $\Sigma^+$ is formed by taking the reciprocal of the non-zero singular values of $A$ and transposing the matrix.

Compact SVD is a variant of SVD where matrices $U$ and $V$ are truncated to include only the columns corresponding to the non-zero singular values, often used when the matrix has more zero singular values.

In SVD, $U$ and $V$ are unitary matrices, meaning they satisfy $UU^* = I$ and $VV^* = I$, where $I$ is the identity matrix and * denotes the conjugate transpose operation.

The equation for SVD of a matrix $A$ is $A = U\Sigma V^*$, where $U$ and $V$ are orthogonal (or unitary, if complex) matrices, $\Sigma$ is a diagonal matrix, and $V^*$ is the conjugate transpose of $V$.

Singular Value Decomposition is a factorization of a matrix into three matrices $U$, $\Sigma$, and $V^*$, where $U$ and $V$ are unitary matrices and $\Sigma$ is a diagonal matrix containing the singular values.

SVD is related to eigen-decomposition in that the singular values of $A$ are the square roots of the eigenvalues of $A^*A$ or $AA^*$, and the columns of $U$ and $V$ are the eigenvectors of $AA^*$ and $A^*A$ respectively.

SVD can be applied to any $m \times n$ rectangular matrix, not just square matrices. The resulting $U$, $\Sigma$, and $V^*$ matrices will have dimensions $m \times m$, $m \times n$, and $n \times n$ respectively in the full SVD case.

SVD is numerically stable, meaning small changes in the input matrix will cause only small changes in the singular values and singular vectors. This makes SVD a reliable method for numerical computations.

Singular values are the non-negative values on the diagonal of matrix $\Sigma$ in the SVD. They are the square roots of the nonzero eigenvalues of $A^*A$ where $A$ is the original matrix.

The rank of a matrix $A$ can be determined using SVD by counting the number of non-zero singular values in $\Sigma$. This corresponds to the number of linearly independent rows or columns in $A$.