What is the actual output of Principal Component Analysis? I'm trying to understand PCA, but I don't have a machine learning background.  I come from software engineering, but the literature I've tried to read so far is hard for me to digest.
As far as I understand PCA, it will take a set of datapoints from an N dimensional space and translate them to an M dimensional space, where N &gt; M.  I don't yet understand what the actual output of PCA is.
For example, take this 5 dimensional input data with values in the range [0,10):
// dimensions:
// a  b  c  d  e

[[ 4, 1, 2, 8, 8],      // component 1
 [ 3, 0, 2, 9, 8],
 [ 4, 0, 0, 9, 1],
 ...
 [ 7, 9, 1, 2, 3],      // component 2
 [ 9, 9, 0, 2, 7],
 [ 7, 8, 1, 0, 0]]

My assumption is that PCA could be used to reduce the data from 5 dimensions to, say, 1 dimension.
Data details:
There are two &quot;components&quot; in the data.

One component has mid a levels, low b and c levels, high d, and nondeterministic e levels.
The other component has high a and b levels, low c and d levels, and nondeterministic e levels.

This means that the two components are most differentiated by b and d, somewhat differentiated by a, and negligibly differentiated by c and e.
Outputs?
I'm making this up, but say the (non-normalized) linear combination with the highest differentiating power is something like
5*a + 10*b + 0*c + 10*d + 0*e

The above input data translated along that single axis is:
[[110],
 [105],
 [110],
 ...etc

Is that linear combination (or a vector describing it) the output of PCA?  Or is the output the actual reduced dataset?  Or something else entirely?
Have you tried reading the Intuition section of the PCA page on wiki? Also, the Further Considerations section, I think, explains what the various components represent and addresses your questions.

A short summary of what those sections describe is that the original components (i.e., axes) of the data do not properly represent the relationship inherit in the data. Instead, the components can be combined to yield a new set of components, or axes, that better describe how the data is distributed.

The wiki link states that "PCA can be thought of as fitting an n-dimensional ellipsoid to the data". In that line of thinking, the output of PCA is the eigenvectors and eigenvalues that define the orientation and length of the ellipsoid that best fits the data. 

What you do with that ellipsoid is up to you and your application of PCA.