What does the singular value decomposition (SVD) express a matrix as?

The singular value decomposition (SVD) is a powerful factorization technique in linear algebra that expresses any given ( m \times n ) matrix ( A ) in terms of three specific matrices: ( U ), ( \Sigma ), and ( V^* ).

In this decomposition:

• ( U ) is an ( m \times m ) orthogonal matrix whose columns are the left singular vectors of ( A ).

• ( \Sigma ) is an ( m \times n ) diagonal matrix that contains the singular values of ( A ) on its diagonal, which are non-negative and are typically arranged in descending order.
• ( V^* ) (the conjugate transpose of ( V )) is an ( n \times n ) orthogonal matrix whose columns are the right singular vectors of ( A ).

Thus, the SVD of a matrix ( A ) can be written as:

[ A = U \Sigma V^* ]

This representation has significant implications in various applications, such as data compression and dimensionality reduction in machine learning, making it a fundamental aspect of applied linear algebra. The structure provided by ( U ), ( \Sigma ), and ( V^