Principal Component Analysis Problem




I have a matrix X of N dimension, where N is the number of the features. Now to reduce the features from N to k (number of principal components), I want to use PCA. But lets say, if I run PCA with k=N, such that the dimension of the data is not reduced at all. After that if we reconstruct the X matrix,which gives us Xapprox. Now will Xapprox be same as my original X, meaning Xapprox = X ?



We can optionally recover the original data back, by 100% if we have chosen all components.
Because taking only few components , is enough to recreate the matrix that has almost all of the features of the original matrix.