Cantor Set

 

 The cantor set cis a subset of the unit interval, with the subspace topology.  Delete (1/3,2/3), then (1/9,2/9) and (7/9,8/9), and so on, removing the middle third of segments forever.

Notice that the section of c from 0 to 1/3, when magnified, looks exactly like c.  This is an early example of a fractal, although fractal geometry was not known at the time.  Cantor simply thought it was a beautiful set, and indeed it is.

If x is in the complement of c, then it was removed at some point, as part of an open interval.  The complement is open, and c is closed.

Being a closed subspace of [0,1], c is compact.

Since c is a closed subspace of a complete metric space, it too is a complete metric space.

Let s be a sequence of zeros and ones.  If s₁ = 0, select the interval [0,1/3].  If s₁ = 1, select the interval [2/3,1].  If s₂ = 0, select the first third of the interval previously selected, and if s₂ = 1, select the last third of the interval previously selected.  This continues forever, building a chain of descending closed intervals whose lengths approach 0.  This chain always converges to a point, which we will call x.  Furthermore, x does not lie in the middle third of any segment, hence x has never been deleted, and x belongs to c.  We have a map from binary sequences into c.

Different sequences will diverge at some point, living in disjoint intervals thereafter.  Thus the map is 1-1.

Finally, let x be a point in the cantor set c.  At each step, x lies in the first or last third of the prior interval.  If it were in the middle third it would not lie in c.  Thus, at each step, s_n can be set to 0 or 1.  The resulting sequence converges to some y in c, and if y is not equal to x, then s_n goes down the wrong path at some point, moving towards y instead of x.  This contradicts the construction of s, hence s defines x.  The map is onto, and the points of c correspond 1-1 with the infinite binary sequences, which are sometimes written 2^ω.

As a corollary, the points of c are uncountable.

The binary sequences can be given an order topology.  This is based on lexicographic order, where 10101... is greater than 10100..., and so on.  Verify that the map from sequences into c respects order.  Thus 2^ω and c are homeomorphic.

 

From Sequences to Reals

It's a little bit off topic, but a similar construction shows ω^ω is homeomorphic to the nonnegative reals.  If s₀ = n then restrict attention to the interval [n,n+1).  If the next integer s₂ = 0, select, from the prior interval, the range [0,1/2).  If s₂ = 1 then select [1/2,3/4).  If s₂ = 2 select [3/4,7/8), and so on.  with this interval established, move to s₃, and select a slice of this interval in exactly the same way.  Continue this process forever, homing in on x.  Technically, each step establishes a half open interval, but you can think of them as closed intervals, giving a descending chain of closed intervals that converges to a point.  As before, this point has to be x.  The map is 1-1 and onto, and it respects order - lexicographic order in the infinite sequences and linear order in the reals.  The two spaces are homeomorphic.

 

trace

trace (linear algebra)

In linear algebra, the trace of an n-by-n square matrix A is defined to be the sum of the elements on the main diagonal (the diagonal from the upper left to the lower right) of A, i.e.

tr(A) = A_1,1 + A_2,2 + ... + A_n,n.

where A_ij represents the (i,j)'th element of A. The use of the term trace arises

 from the German term Spur (cognate with the English spoor).

Properties

The trace is a linear map. That is,

tr(A + B) = tr(A) + tr(B)

tr(rA) = r tr(A)

for all square matrices A and B, and all scalars r.

Since the principal diagonal is not moved on transposition, a matrix and its transpose have the same trace:

tr(A) = tr(A^T).

If A is an n×m matrix and B is an m×n matrix, then

tr(AB) = tr(BA).

Note here that AB is an n×n matrix, while BA is an m×m matrix.

Using this fact, we can deduce that the trace of a product of square matrices is equal to the trace of any cyclic permutation of the product, a fact known as the cyclic property of the trace. For example, with three square matrices A, B, and C,

tr(ABC) = tr(CAB) = tr(BCA).

More generally, the same is true if the matrices are not assumed to be square, but are so shaped such that all of these products exist.

If A, B, and C are square matrices of the same dimension and are symmetric, then the traces of their products are invariant not only under cyclic permutations but under all permutations, i.e.,

tr(ABC) = tr(CAB) = tr(BCA) = tr(BAC) = tr(CBA) = tr(ACB).

The trace is similarity-invariant, which means that A and P⁻¹AP (P invertible)

 have the same trace, though there exist matrices which have the same trace but are not similar. This can be verified using the cyclic property above:

tr(P⁻¹AP) = tr(PP⁻¹A) = tr(A)

Given some linear map f : V → V (V is a finite-dimensional vector space) generally, we can define the trace of this map by considering the trace of matrix representation of f, that is, choosing a basis for V and describing f as a matrix relative to this basis, and taking the trace of this square matrix. The result will not depend on the basis chosen, since different bases will give rise to similar matrices, allowing for the possibility of a basis independent definition for the trace of a linear map. Using the canonical isomorphism between the space End(V) of linear maps on V and V⊗V^*, the trace of v⊗f is defined to be f(v), with v in V and f an element of the dual space V^*.

Eigenvalue relationships

If A is a square n-by-n matrix with real or complex entries and if λ₁,...,λ_n are

 the (complex) eigenvalues of A (listed according to their algebraic multiplicities), then

tr(A) = ∑ λ_i.

This follows from the fact that A is always similar to its Jordan form, an upper triangular matrix having λ₁,...,λ_n on the main diagonal.

From the connection between the trace and the eigenvalues, one can derive a connection between the trace function, the matrix exponential function, and the determinant:

det(exp(A)) = exp(tr(A)).

The trace also prominently appears in Jacobi's formula for the derivative of the determinant (see under determinant).

Other ideas and applications

If one imagines that the matrix A describes a water flow, in the sense that for every x in Rⁿ, the vector Ax represents the velocity of the water at the location x, then the trace of A can be interpreted as follows: given any region U in Rⁿ,

the net flow of water out of U is given by tr(A)· vol(U), where vol(U) is the volume of U. See divergence.

The trace is used to define characters of group representations. Given two representations A(x) and B(x), they are equivalent if tr A(x) = tr B(x).

The trace also plays a central role in the distribution of quadratic forms.

A matrix whose trace is zero is said to be traceless or tracefree.

Inner product

For an m-by-n matrix A with complex (or real) entries, we have

tr(A^*A) ≥ 0

with equality only if A = 0. The assignment

= tr(A^*B)

yields an inner product on the space of all complex (or real) m-by-n matrices.

If m=n then the norm induced by the above inner product is called the Frobenius norm of a square matrix. Indeed it is simply the Euclidean norm if the matrix is considered as a vector of length n².