Define the singular vectors of a matrix.

The singular vectors of a matrix are the eigenvectors of the matrix multiplied by its transpose.

Singular vectors are a key concept in linear algebra and are used in a variety of applications, including data analysis and image processing. They are defined as the eigenvectors of a matrix multiplied by its transpose. Specifically, if A is an m x n matrix, then its singular vectors are the eigenvectors of the matrix A^T A or AA^T. These vectors are orthogonal and form the basis for the left and right singular subspaces of A.

The singular vectors of a matrix are important because they provide a way to decompose the matrix into its constituent parts. This is known as the singular value decomposition (SVD) and is a powerful tool in linear algebra. The SVD of a matrix A is given by A = UΣV^T, where U and V are the left and right singular vectors of A, respectively, and Σ is a diagonal matrix containing the singular values of A.

The singular vectors of a matrix can also be used to solve linear systems of equations and to perform principal component analysis (PCA) on data sets. In PCA, the singular vectors of the covariance matrix of the data set are used to identify the most important features of the data and to reduce its dimensionality.

In summary, the singular vectors of a matrix are the eigenvectors of the matrix multiplied by its transpose and are important in a variety of applications in linear algebra, data analysis, and image processing. They provide a way to decompose a matrix into its constituent parts and can be used to solve linear systems of equations and perform PCA on data sets.