12 Techniques: Factor Decomposition of Data Matrices

Question 1

What is row space?

Answer 1

the set of all possible linear combinations of the p-dimensional row vectors of a data matrix Xₙₓₚ

Question 2

What is column space?

Answer 2

the set of all possible linear combinations of the n-dimensional column vectors of a data matrix Xₙₓₚ

Question 3

What is the idea behind row and column space?

Answer 3

• approximating row and column spaces with smaller subspaces
• preserving the data structure and variation to a large degree

Question 4

What is unit vector u₁*?

Answer 4

• gives the direction of a line through the origin

- optimizing initial cloud of n points (least squares)

Question 5

What is the solution for row factor decomposition? The corresponding factor?

Answer 5

• solution: eigenvector associated with the qth largest eigenvalue
• qth factor : zₖ* = X ⋅ uₖ* (Note: q = k)

Question 6

What is the solution for column factor decomposition? The corresponding factor?

Answer 6

• solution: eigenvector associated with the qth largest eigenvalue
• qth factor : wₖ* = Xᵀ ⋅ vₖ* (Note: q = k)

Question 7

When asked to discuss factor decomposition of data matrix X, what do you first do?

Answer 7

find XᵀX and XXᵀ