Recent content by mind0nmath

Homework Statement boy and girl independently flip each a biased coin with probability of heads p1 for boy, and p2 for girl. they record the number of flips needed until heads shows up. what is the probability that they tie? Homework Equations I just know that we should define a...

if A* is the adjoint of A in Complx, Then is A* x A* = (A^2)* or something else??

Here's a new question(related to the subject). Is the 2x2 matrix with only a scalar in the right upper corner and all other entries = 0, diagonalizable? That matrix, call it T, is nilpotent. T^2 is the zero matrix. Isn't the zero matrix diagonalizable? it is in the diagonal form so it must be...

How about the addition of two diagonalizable matricies? Is it always diagonalizable in some basis if the field is the complex numbers? my initial thought is: yes. by the spectral theorem. but not 100% sure since I suspect we might not be able to find a basis for the addition of the matrices...

I thought for T to be diagonalizable, it should have dimT distinct eigenvalues or dim(T) linearly independent eigenvectors. Then how is the Zero matrix diagonalizable? aren't the roots all the same(i.e. =0)? and null(T)=C^n?

So if T is over the complex numbers, then both directions are true? i.e. T is diagonalizable <=> T^2 is diagonalizable?

Hey, I know that if T(linear transformation in finite dim-vector space) is diagonalizable, then the matrix A that represent T is diagonalizable if there exist a matrix P=! 0 that is invertible and A=P^-1 * D * P for a diagonal matrix D. I also know that raising A to a positive power will...

got it. thanks.

Hey, Is the generalized eigenspace invariant under the operator T? Let T be finite dimensional Linear operator on C(complex numbers). My understanding of the Generalized Eigenspace for the eigenvalue y is: "All v in V such that there exists a j>=1, (T-yIdenitity)^j (v) = 0." plus 0. thanks

Hey. What can be said about all the normal nxn matrices that have exactly 1 eigenvalue? I'm interested in the case where the entries are in C (complex #'s). what sort of generalizations can we make? thanks.

Hey, I know how to find the expected value from the density function when it is in the form: (example) | y^2 -1<y<1 | fy =| | 0 elsewhere Ey = integral(upper limit 1, lower limit -1)[y*y^2 dy) but, what if the density function looks like this...

Hey, I know how to find the expected value from the density function when it is in the form: | fy =

isn't it true that all linear operators on a finite dimensional vector space have eigenvalues? I don't really know what [S, T] is though.

how about for something like: T(x_1,x_2,...,x_n) = (x_1+x_2+...+x_n, x_1+x_2+...+x_n, ..., x_1+x_2+...+x_n). The matrix with respect to standard basis would have 1's everywhere? any clues to finding the eigenvalues/vectors?

thanks but I think I have the answer. If you pick an arbitrary vector in V and define T(u) = au and S(u) = bu, where a,b are eigenvalues for u, then applying S to T(u) and T to S(u) will give the wanted results.