Proposition (Simplification of the Norm)

In transform coding we can simplify the distortion term by acknowledging that the following equality holds $$\lVert \mathbf{X}-\mathbf{\hat{X}} \rVert =\lVert \mathbf{Y}-\mathbf{\hat{Y}} \rVert$$meaning we can simplify our distortion to $$D_{\text{tc}}=\sum_{i=1}^{k}E[(Y_{i}-\hat{Y}_{i})^{2}]=E[\lVert \mathbf{Y}-\mathbf{\hat{Y}} \rVert ^{2}]$$