Recent content by MikeDietrich

I don't know if A^2 will be positive definite. Oh, wait... if A^2= SD^2S^-1 then the diagonal matrix squared will only give positive or zero eigenvalues so A^2 will be positive semidefinite unless A is invertible then it would be positive definite. Do I have to worry about the orthogonal...

Homework Statement If A is a symmetric matrix, what can you say about the definiteness of A^2? Explain. Homework Equations I believe I need to use the face that A^2=SD^2S^-1. I know that if all the eigenvalues of a symmetric matrix are positive then the matrix is positive...

Homework Statement This is a general question... I can easily go from a matrix A to its eigenvalues and then eigenvectors but how would I go from the eigenvalues and eigenvectors to a feasible original matrix? Any thoughts appreciated!

Homework Statement True or False and Why? "The sum of two quadratic forms in three variables must be a quadratic form as well." Homework Equations q(x_1,x_2,x_3)=x_1^2+x_2^2+x_3^2+x_1x_3+x_2x_3 The Attempt at a Solution I am definitely missing something. To me this is a...

I know I also need to show injectivity and surjectivity (I assume this is what you are getting at). In my attempt at brevity I was not clear and I thank you for your patience.

5/4(n) is the homomorphism that would map 4Z to 5Z (yes? no?). Maybe I'm confused... I did not think/realize/know the a homomorphism needed to be in the set?

Homework Statement The question asks me to determine if 4Z and 5Z (with standard addition) are isomorphic and if so to give the isomorphism. 2. The attempt at a solution What I am having difficulty with is showing a mapping that preserves the operation. IE phi(a+b) = phi(a) + phi(b)...

Homework Statement Draw two plane figures, each having a 12 element group of symmetries, such that the two groups are NOT isomorphic. Demonstrate that they are not isomorphic.Homework Equations I know that every finite group of isometries of the plane is isomorphic to either Z_n or to the...

Homework Statement Do the following have the same cardinality? If so, establish a bijection and if not explain why. A line segment of 4 units and half of a circumference of radius 1 (including both endpoints). The attempt at a solution So my thought is that if I manipulate the...

Thank you. That is exactly what I was looking to know!

sorry... I meant "Abelian" not "non-Abelian" (for some reason, I cannot edit my original post). Thank you Mark.

Homework Statement The problem asks me to determine if the matrix [p -q ## q p] is a field with addition and multiplication. However, that is not my question. My question is: How is proving a set is a field different from proving a set is a non-abelian group (under addition then separately...

Yeah... you're right. I was over-thinking this one. THANKS!

Sure. r = rotation, s = refection Let's say r_1=3, r_2=1, r_3 = 2 and s_1 = 2, s_2 = 1 and s_3 = 3 then (r)(s) = (1,3,2) and (s)(r) = (3,2,1). Obviously, they do not commute. How do I generalize that to include all n>=3?

Homework Statement Prove that S_n and D_n for n>=3 are non-cyclic and non-abelian. Homework Equations I get that I need to show that two elements from each group do not commute and that there is not a single generator to produce the groups... I am just unsure of how to do this...