# Symmetric low rank approximate factorization of symmetric matrix

I am reading a paper and basically there is a covariance matrix (symmetric and positive semi-definite by property) $$V \in R^{d \times d}$$ and we want to write $$V \approx QQ^T$$ with $$Q \in R^{d \times a}, a \ll d$$ It is written we can do this by truncated SVD. I am a little unsure whether I understand this correctly, can someone please verify the steps I think is required to make the approximation?

Firstly, I know we can write $$V = B \Lambda B^T$$ where $$B$$ is an orthogonal basis given by the eigenvectors(+ Gram Schmidt Orthogonalization) and $$\Lambda$$ is a diagonal matrix consisting of the eigenvalues of $$V$$. Rewrite it as $$V = CC^T$$ where $$C = B\sqrt\Lambda$$.
Next we find SVD, $$C = U\Sigma V$$ and make an approximation by truncating the SVD to first $$a$$ values and get a matrix $$D \approx C$$. Finally, $$V \approx D D^T$$.
However, there is one problem in this approach I think. Matrix $$D$$ obtained is $$d \times d$$ and not $$d \times a$$ as required. Am I doing something wrong?

Note: The words exactly used in the paper was - symmetric low rank approximation factorization

Once you have $$V=B\Lambda B^T$$, this is an SVD decomposition of $$V$$.
We assume that the diagonal entries are sorted in non-increasing manner. Let $$B=[B_1, B_2]$$ where $$B_1$$ has $$a$$ columns and $$\Lambda = \begin{bmatrix} \Lambda_1 & 0 \\ 0 & \Lambda_2\end{bmatrix}$$ where $$\Lambda_1 \in \mathbb{R}^{a \times a}$$, then
$$Q=B_1\sqrt{\Lambda_1} \in \mathbb{R}^{d \times a}$$