Question

According to the property of Singular Value Decomposition , it is always possible to decompose a real matrix A into $$ A = U \sum V^T . true or false

Answer 1

According to the property of Singular Value Decomposition ,

It is always possible to decompose a real matrix A into A = $U\sum V^T$. This is a true statement.

As, Suppose A is a $m \times n$ matrix, whose entries come from the field K, where K is either field of real or complex numbers.

Then there exists a factorization called Singular value decomposition of the form

$U\sum V^T$,

where U is a $m \times m$ unitary matrix over K,  $\sum$  is a diagonal $m \times n$ matrix with non-negative real numbers on the diagonal, V is a $n \times n$unitary matrix over K and $V^T$ is the conjugate transpose of V.