Statements about the SVD of a matrix.

Question

Given a matrix $A \in \mathbb{R^{m \times n}}$ is some matrix with the single value decomposition of $A = U \Sigma V^T$, where $U$ contains the left orthogonal singular vectors and $V$ contains the right orthogonal singular vectors and $\Sigma$ contains the singular values in the diagonal, which of the following is true?

1. All singular values in $\Sigma \geq0$.
2. If $A$ is a square matrix that is not invertible, then there exists at least one singular value where $\delta_i=0$
3. If $A$ is square and singular, then all singular values are $\delta_i=0$.
4. The right singular vectors correspond to the the eigen vectors of the matrix $A^TA$.

I know that 1. is true because when we get $\delta_i = ± \sqrt{\lambda_i}$, we always take the positive values and thus all singular values have to be bigger than $0$.

For 4. I'm not sure but I think it's false because if we consider $$A^TA ≡ (U\Sigma V^T)^T(U \Sigma V^T)≡ V\Sigma^TU^TU\Sigma V^T≡V\Sigma^2V$$ which means that $A^TA$ corresponds to $V\Sigma^2V^T$ but I'm not sure how to go on from here.

For 2. and 3. I'm not sure how I can confirm or deny the statements.

Answer 1

$(1)$ is good.

For $(2)$, note that asking about the invertibility of $A$ is the same as asking about the invertibility of $\Sigma$ (as $U$ and $V$ are orthogonal). Since $\Sigma$ is diagonal, it is non-invertible iff one of the diagonal entries is $0$ i.e. if there is a zero singular value.

For $(3)$, what if you looked at a matrix like $\begin{bmatrix} 1 & 0\\ 0 & 0\end{bmatrix}?$

For $(4)$, you should look at the definition of right singular vector. Also, you've shown that $A^TA=V\Sigma^2V^T,$ and you known that $\Sigma^2$ is a diagonal matrix filled with the eigenvalues of $A^TA.$ What does that tell you about $V?$

Answer 2

2. If $A$ is square and no singular value is zero, define $\Sigma^+$ as $\Sigma^+_{ii}=1/\Sigma_{ii}$ and $A^+=V\Sigma^+U^T$ . Then what is $AA^+$?

3. Consider $$ \left( \begin{matrix} 1 & 0 \\ 0 & 0 \end{matrix} \right). $$

4. How do you define eigenvalue decomposition? Is what you obtained the eigenvalue decomposition of $A^TA$?