Is the inverse of a symmetric matrix also symmetric? All the proofs here use algebraic manipulations But I think it may be more illuminating to think of a symmetric matrix as representing an operator consisting of a rotation, an anisotropic scaling and a rotation back This is provided by the Spectral theorem, which says that any symmetric matrix is diagonalizable by an orthogonal matrix With this insight, it is easy to see that the inverse of
What is conjugate in group theory? - Mathematics Stack Exchange Definition of conjugates: Conjugate wiki link Suppose G is a group Two elements a and b of G are called conjugate if there exists an element g in G with g ∗ a ∗ g − 1 = b Here ∗ is operation on group Question: If g ∗ g − 1 will give us identity Then equation will become a ∗ e = b hence a = b Then why we call them conjugate of each other? I am missing basic thing here
abstract algebra - $x$ conjugate to $y$ in a group $G$ is an . . . I am writing this answer largely to push through that, Group theory is not a collection of discrete facts, but really a continuum of ideas about symmetry in Mathematics So, having solved this exercise, you can ask yourself the following questions What is the number of equivalence classes (or since the relation is conjugation, conjugacy classes)? Answer This may not have a nice answer for
Does $\\pi$ contain all possible number combinations? π Pi Pi is an infinite, nonrepeating (sic) decimal - meaning that every possible number combination exists somewhere in pi Converted into ASCII text, somewhere in that infinite string of digits is the name of every person you will ever love, the date, time and manner of your death, and the answers to all the great questions of the universe Is this true? Does it make any sense ?
How to prove $\\text{Rank}(AB)\\leq \\min(\\text{Rank}(A), \\text{Rank . . . The rank of a matrix A which maps V to W is the same as the dimension of the image of V, which will be some subspace U AB is the composite of two linear maps Try tracking the image and its dimension through subspaces as you do one after the other