Recent content by onako

Given two systems, Ax=b and Cy=d, for nxn matrices A and C, and n-dimensional vectors b and d, each of which has at least one solution, it is know that one solution is common to both (satisfies both equations). Could such solution be z found by solving Az+Cz=b+d? I understand that a common...

So, adding a(x-y) to x, means that the distance x to y changes depending on a: positive a implies increased distance, and negative a implies decreased distance? It's a bit "non-rigid" to state that a vector is added to a point.

Suppose that points x and y are given in Euclidean space. Point x is displaced to point x1 by x1=x+a(x-y) Given that a is positive number, how can it be shown that the distance x1 to y is larger than distance x to y. I'm mainly interested in a vector interpretation of the above update...

I'm concerned with a 2D case, with a straight line.

Let's simplify the question. Suppose that non-negative arbitrary weights o_i are associated with each x_i chosen from the straight line. Does x_o lie on that same line? The point is: points x_i are chosen as arbitrary points from the line, and are associated coefficients o_i which are non-negative.

Suppose a set of k arbitrary points, x_i, 1<=i<=k, x_i from R^2 are selected from a line. How can it be shown that a weighted barycenter x_o=(o_i*x_i)/(o_1+o_2+...+o_k) also belongs to that line (assume o_i are arbitrary weights)? Does the choice of weights restrict the solutions (ie, a...

@mathwonk Thanks. What is the exact meaning of "at least" in your statement? Could the set of vectors of null vectors be a superset of the set of eigenvectors corresponding to zero eigenvalue (of course, any linear comb of such eigenvectors will also be a vector of null space)?

Suppose a square matrix A is given. Is it true that the null space of A corresponds to eigenvectors of A being associated with its zero eigenvalue? I'm a bit confused with the terms 'algebraic and geometric multiplicity' of eigenvalues related to the previous statement? How does this affect the...

It then means you're squaring each term, and not the function itself. If a function is squared, then these would be equivalent. Given a set of points in 2D, a point that minimizes the sum of squared distances to such points is the barycenter; I'm not sure about the sum of distances (so, not...

Thanks. Now, faced with the problem of minimizing f(x) for provided 2D parameters x1, x2, x3, ..., x_k, one sets the derivative to zero, and computes for x. However, in case of more than one dimension this problem is non-trivial, I think. What would be the minimizer of f(x), provided 2D...

Provided is a function f(x)=\sum_{j=1}^n ||x-x_j||, for x being a two dimensional vector, where ||.|| denotes the Euclidean distance in 2D space. How could one obtain a derivative of such a function?

Thank your for such a good explanation.

Thanks. Just one note: I suppose you've taken into account that there are p columns in X (which is an n x p matrix). If I'm not wrong, only n linearly independent columns of dimensionality R^n define a basis in R^n. So, given an input X, with linearly independent columns, such columns could...

Suppose a matrix X of size n x p is given, n>p, with p linearly independent columns. Can it be guaranteed that there exists a matrix A of size p x p that converts columns of X to orthonormal columns. In other words, is there an A, such that Y=XA, and Y^TY=I, where I is an p x p identity matrix.

Could you provide a simple example?