Quiz on Singular Value Decomposition

Every real $m \times n$ matrix $A$ can be written as $A = U \Sigma V^T$, where $U$ and $V$ are orthogonal matrices and $\Sigma$ is diagonal with nonnegative entries.

The nonzero singular values of a matrix $A$ are the square roots of the nonzero eigenvalues of both $A^T A$ and $A A^T$.

In the SVD $A = U \Sigma V^T$, the columns of $U$ form an orthonormal basis for the column space of $A$, and the columns of $V$ form an orthonormal basis for the row space of $A$.

If $A$ is an $m \times n$ matrix of rank $r$, then in its SVD $A = U \Sigma V^T$, the number of nonzero singular values is always $m + n – r$.

Truncating the SVD by keeping only the $k$ largest singular values gives the best rank-$k$ approximation to a matrix $A$ in the Frobenius norm sense.

If $A$ is a square orthogonal matrix, then all of its singular values are equal to $1$.

The SVD of a matrix $A$ is unique: there is only one possible triple $(U, \Sigma, V)$ that satisfies $A = U \Sigma V^T$.