Can the Eigen faces be negative?

I have checked several sites and found that eigen faces are Eigen Vectors. PCA transforms the faces into a new space such that the hyper plane is in the direction of maximum variance. I have attached a link which shows green and red arrows and some holes: Could you explain Eigen Faces

and the holes I think represent the Eigen faces. So red arrow represents the principal axis. So its possible to see that some of the holes which have negative values can project on to the principal axes. But when we say that Eigen faces are vectors then their direction can be negative but magnitude cannot. Please guide me whether Eigen faces can have negative values? Zulfi.

• Are you confusing Eigen faces with eigenvalues? It is the latter that cannot be negative (unless your staying matrix is not positive semi definite). Magnitude by definition is directionless, it is size regardless of direction. Nov 8, 2018 at 7:41

In PCA, the face images are formed from a linear combination of the eigenfaces, using both positive/negative weights.

The "eigenfaces" arise in facial recognition problems using principal component analysis (PCA). They are the eigenvectors of the sample covariance matrix of the initial face images. As with any application of PCA, the original data vectors can be represented as a linear combination of the eigenvectors of their covariance matrix --- that is, each of the faces in the original data can be obtained as a linear combination of the eigenfaces. More importantly, we can approximate faces well by taking only a truncated set of the most important eigenfaces (i.e., the ones corresponding to the highest eigenvalues).

When we represents faces in the original data as a linear combination of the eigenfaces, the elements of this linear combination can have positive or negative coefficients. So yes, a face in the original data may be represented with a negative value associated with an eigenface (i.e., it is formed by subtracting some amount of that eigenface from other eigenfaces).

The graph in the link in your question is simply showing the principal components of data point in two dimensions on a graph. The circles in this graph represent data points (not holes) and the red and green arrows show the first and second principal components of the data. So in this graph, the red and green arrows would be the "eigenfaces" (for face images with only two pixels) and each of the data points can be represented as a linear combination of these "eigenfaces". When you represent the data points in PCA you are looking at them the data on this new set of axes. So, taking the red axis as an example, you can see that some of the points will have positive values on this axis, and some will have negative values.

• Thanks for clarifying the things.So what I understand is that : Eigen faces cannot be negative, eigen values can be negative and the circles in the image represents the data points associated with the Eigen faces. An Eigen face in the original data can be negative but an Eigen face in the new space cannot be negative. But still I am confused: Even in the new space why eigen vectors can't have negative coefficients? Zulfi. Nov 8, 2018 at 16:14
• Eigenfaces are vectors in a high-dimensional space, so what would "negative" mean there? They can certainly be negative on some axis determined by another vector, if that's what you mean.
– Ben
Nov 8, 2018 at 22:43
• Thanks. God bless you for explaining me this. By negative, I mean that they would be in opposite direction of positive Eigen Vectors. Because we do have origin in higher dimensional space. Nov 9, 2018 at 23:21