What is the optimal point in the nullspace of A using Lagrange multipliers?

• hoffmann
In summary, the point x* in the nullspace of A is significant because it is the point closest to the origin (x0) of the nullspace. This point is also the solution to the linear system Ax* = 0.
hoffmann
Does anyone know how to approach this problem?

Let A be an m×n matrix of rank m, where m<n. Pick a point x in R^n, and let x∗ be the point in the nullspace of A closest to x. Write a formula for x∗ in terms of x and A.

What exactly is the significance of the point x* in the nullspace of A?

i think this question has something to do with the orthogonality of subspaces and is analogous to saying the shortest distance between a point and an axis is just a perpendicular line...

>> What exactly is the significance of the point x* in the nullspace of A?

So, Ax* = 0 which is basically what it means by x* being in the nullspace of A. x* is a vector which multiplies A to take it to 0.

I think it has something to do with the least square solution for linear systems.

I think you should think of the solution in terms of the projection of x onto the nullspace of A, which would be x*.

This is simply given by A*(A'A)-1*A'*x. Sorry, I am not fluent with latex.

A' = transpose of A.
-1 = inverse
* = multiplication

Refer to the Strang book on least square and projection of vectors onto the column space of a a matrix.

We know that the row space is in $$R^{n}$$ and that it is orthogonal to the null space.

Imagine that we have a 2x3 matrix with rank 2. It's row space would be a plane in $$R^{3}$$, and it's null space a line perpendicular to that plane. If we pick a point $$x$$ in that plane, wouldn't the point closest to the null space be that same point x?

I'm eagerly awaiting someone to help us out :)

^^ i think you're on the right track, dafe.

i think an analogy to this problem is, what is the shortest distance between a point in the xy plane and the x axis. it's just a line perpendicular to the x-axis.

so in the context of the problem i asked, it's like...the dot product between the point x and...the row space? something like that...

ok, some help:

1) set this up with a variable (btw, better to call the starting point x0 rather than x, since it is fixed), an objective function and a set of equality constraints. Write them down. You should replace the objective function by something nicer: for example no square roots, since to apply Lagrange you need a differentiable function.
2) show geometrically in terms of the level sets of the objective function and the tangent space of the constraints that the vector x0 - x* is perpendicular to the null space.
3) form the Lagrangian. solve.

What is the nullspace of an mxn matrix?

The nullspace of an mxn matrix is the set of all vectors that, when multiplied by the matrix, result in a zero vector. In other words, it is the set of all solutions to the equation Ax = 0, where A is the mxn matrix and x is a vector with n components.

How is the nullspace of an mxn matrix related to its rank?

The nullspace of an mxn matrix has a dimension of n - r, where r is the rank of the matrix. This means that the number of linearly independent vectors in the nullspace is equal to the number of columns in the matrix minus its rank.

What does the nullspace of an mxn matrix tell us about the matrix?

The nullspace of an mxn matrix can tell us about the linear dependence of its columns. If the nullspace contains only the zero vector, then the columns of the matrix are linearly independent. If the nullspace contains other vectors, then the columns are linearly dependent.

How can we find the nullspace of an mxn matrix?

The nullspace of an mxn matrix can be found by performing row reduction on the matrix and identifying the free variables. The columns corresponding to the free variables will form a basis for the nullspace.

What is the significance of the nullspace of an mxn matrix?

The nullspace of an mxn matrix is important in understanding the solutions to systems of linear equations. It can also be used in applications such as data compression and solving optimization problems.

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