Suppose A is an m*n matrix with real values. It has a Null Space N(A) and a rank r. Can we infer N(ATA) and its rank?
We know that N(A) is contained in N(ATA), because if Ax = 0 then ATAx = 0. But how can we be sure that no x exists such that Ax != 0 but ATAx = 0?
Ax is a combinations of the columns of A, so it belongs to the columns space of A ( C(A) ) or equivalently to the row space of AT. At the same time, if AT(Ax) = 0, then it means that Ax belongs to the null space of AT. But we know that these 2 vector subspaces are orthogonal and share only the 0 vector; otherwise it would be that (Ax)T(Ax) = 0 while Ax != 0, but the inner product of a real vector is the square of its length, so it cannot be 0 for a non zero vector!
And this demonstrate that N(ATA) = N(A).
Because the rank of a matrix m*n is equal to n – dimension of N(matrix), we can also say that the rank(ATA) = rank(A).
I would like to point out a very nice course on general relativity, I think the best you can find between video courses on the web. It’s title is: “International Winter School on Gravity and Light 2015” and you can find it on YouTube at the link: gravity and light
The course is taught by Professor Frederic Schuller, who has a deep understanding of the subject. It is a real pleasure to follow the logical construction that starts from the definition of what a topology is and gradually introduces more and more complex concepts: all explained always with a strong mathematical rigor but, at the same time, with great clarity.
I have found some difficulties in relation to the Affine Connection, so I started looking for articles or books that could help be on the subject. Thanks to the suggestion of Professor Edmund Bertschinger in his article, I bought the book “Geometrical methods of mathematical physics” by Bernard Schutz, which is written very well and deals with the Affine Connection in its last Chapter. I’ve not already read that chapter, but I trust that it will be treated clearly and comprehensively!
In this book some basis of algebra are required, as during the presentation of the metric tensor. If you need to refresh or learn from scratch the topic, I suggest a very nice course on MIT platform OpenCourseWare: Linear Algebra by Professor Gilbert Strang.
He does not demonstrate many passages but provides the student with many examples that allow him to find them for himself.