# Question about PCA data recovery equation

In PCA , consider a 4 x 3 data matrix ( 4 examples each with 3 features ). After getting the 3 eigenvectors (a/b/c) and projecting data on the first 2 vectors, the equation looks like this :

[ first 2 eigen vectors transposed ] x [ data set transposed ] = [ data projected ]

so the first example in the data after being projected would consist of:

$$[a_1x_1 + a_2x_2 + a_3x_3]\\ [b_1x_1 + b_2x_2 + b_3x_3]$$

On recovering data the equation looks like this:

[first 2 eigen vectors] x [projected data] = [original data set]

so that the first example consists of:

$$a_1[a_1x_1 + a_2x_2 + a_3x_3] + b_1[b_1x_1 + b_2x_2 + b_3x_3]\\ a_2[a_1x_1 + a_2x_2 + a_3x_3] + b_2[b_1x_1 + b_2x_2 + b_3x_3]\\ a_3[a_1x_1 + a_2x_2 + a_3x_3] + b_3[b_1x_1 + b_2x_2 + b_3x_3]$$ My question is how is this equivalent to what the original example looked like $[x_1 x_2 x_3]'$? I know that unit vectors will cancel each other but the equation will be like

$$a_1[a_1x_1] + b_1[b_1x_1]\\ a_2[a_2x_2] + b_2[b_2x_2]\\ a_3[a_3x_3] + b_3[b_3x_3]$$

giving : $$x_1 + x_1\\ x_2 + x_2\\ x_3 + x_3$$

If $V$ is the matrix whose columns are the eigenvectors and $x$ is a 3-element column vector, then the transformed vector is given by
$$x' = V^{T}x$$
$$x = Vx'= V[V^{T}x] = V[V^{-1}x] = [VV^{-1}]x = x$$
But note that in your example, since you only projected $x$ onto the first two eigenvectors, you can not, in general, recover the original vector. You would need to project $x$ onto the full set of eigenvectors for it to be reversible (assuming there are no redundant eigenvectors).