struggling on the properties of the idempotent matrix, namely for any n &#x00D7;<!-- × --> n

protestommb 2022-06-24 Answered
struggling on the properties of the idempotent matrix, namely for any n × n matrix, A 2 = A .. The projection matrix defined by M = I n A ( A T A ) 1 A T is an idempotent matrix. The question is, for any given n × m ( n > m) matrix B ,, do we have
M = I n A ( A T A ) 1 A T = I n A B ( B T A T A B ) 1 B T A T ,
since A B is basically the linear transformation of matrix A .
You can still ask an expert for help

Want to know more about Matrix transformations?

Expert Community at Your Service

  • Live experts 24/7
  • Questions are typically answered in as fast as 30 minutes
  • Personalized clear answers
Learn more

Solve your problem for the price of one coffee

  • Available 24/7
  • Math expert for every subject
  • Pay only if we can solve it
Ask Question

Answers (1)

Daniel Valdez
Answered 2022-06-25 Author has 19 answers
First, your projection matrix makes sense if r a n k A = k < n. Otherwise, M = 0. If so, M maps to the n k-dimensional linear space spanned by the vectors c i orthogonal to the columns of A.
Now, when you consider M ¯ = I n A B ( B T A T A B ) 1 B T A T , this operator maps to the linear subspace orthogonal to the column space of A B. Here, you can have different options depending on the rank of A B.
Let, for instance, r a n k A B = r a n k A. In this case, the column space of A B is the same as the column space of A and M = M ¯ . But this will not be the case if r a n k A B < r a n k A.
Not exactly what you’re looking for?
Ask My Question

Expert Community at Your Service

  • Live experts 24/7
  • Questions are typically answered in as fast as 30 minutes
  • Personalized clear answers
Learn more

You might be interested in

asked 2021-09-18

Find an explicit description of Nul A by listing vectors that span the null space.
A=[154310121000000]

asked 2021-06-13
For the matrix A below, find a nonzero vector in Nul A and a nonzero vector in Col A.
A=[2359891121431727]
Find a nonzero vector in Nul A.
A=[3201]
asked 2021-09-13

Assume that A is row equivalent to B. Find bases for Nul A and Col A.
A=[12511324515212045365192]
B=[12045005780000900000]

asked 2021-09-30
The matrix in Hessenberg form with the help of similarity transformation and the matrix in the similarity transformations.
asked 2022-01-20
Matrix of singular transformation
A=(0,1) to A=(0,0), B=(2,0) to B=(0,1) and C=(2,1) to C=(0,2)
find the matrix representation r?
asked 2021-12-19
How can I find the dimension of the eigenspace?
The matrix A=[9117] has one eigenvalue of multiplicity 2. Find this eigenvalue and the dimension of the eigenspace.
So we found the eigenvalue by doing AλI to get:
λ=8
But how exactly do we find the dimension of the eigenspace?
asked 2022-06-24
struggling on the properties of the idempotent matrix, namely for any n × n matrix, A 2 = A .. The projection matrix defined by M = I n A ( A T A ) 1 A T is an idempotent matrix. The question is, for any given n × m ( n > m) matrix B ,, do we have
M = I n A ( A T A ) 1 A T = I n A B ( B T A T A B ) 1 B T A T ,
since A B is basically the linear transformation of matrix A .

New questions