Projection into the left null space

Homework Statement

I am trying to find the matrix M that projects a vector b into the left nullspace of A, aka the nullspace of A transpose.

Homework Equations

A = matrix
A ^ T = A transpose
A ^ -1 = inverse of A
e = b - A x (hat)
e = b-p

I know that the matrix P that projects the vector b into the collumn space of A is P = A(A ^T*A)^-1 A^T. Col space is orthogonal to the left nullspace

The Attempt at a Solution

Since Col space is orth to left null, I was thinking of just find a matrix that, when doted with P is equal to zero (the definition of orthogonality); but thats what they want us to do in part b

Also, since we can get the left nullspace from the column space, i was thinking we could just apply that to P in order to get M (as in find the left null space of P) but the problem is that A is not given

Third idea; the error e used in finding P is in the left nullspace. so if i could somehow make it only have a component in the left nullspace, none in the column space, i could somehow find P.

So i have plenty of ideas, but no idea how to implement them. any help would be GREATLY appreciated, as this pset is due in 3 hours!

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heh, it is actually very simple... and you have basically solved it... yes use that error term...

a vector consists of components in Col A and components in Col A perp... so v=x+y, x in Col A and y in Col A prep. just subtract y from x and there you go!

x=Px + y, so y=x-Px or just (I-P)x

edit: fixed the typo.

Last edited:
AKG
Homework Helper
This doesn't look right. If A has a non-trivial nullspace, then (ATA)-1 doesn't exist, so your formula for the matrix which projects to column space of A doesn't make sense when A is singular. On the other hand, if A is non-singular, then A(ATA)-1AT = I, so your formula just gives the identity function, but it's obvious that the identity function is what projects the column space of A when A is non-singular.

I think he meant that A is the matrix whose columns span a vector space. thus A is nonsingular (though not necessarily a square matrix). the P matrix is the projection operator of a vector onto Col A. of course, if A is a square matrix.. there is nothing to project, thus P becomes identity.

edit: sry I meant the columns of A consists of a set of basis for Col A.

Last edited:
AKG