write the exchange of D2
Answers
Answer:
hi mate
your answer is D2 = 406
Explanation:
f I understand the question correctly, you work in the space of real 29×29 matrices and you have to compute the respective dimensions of the subspaces which are given by matrices that are
1) diagonal
2) upper triangular
3) trace-zero
4) symmetric
5) skew-symmetric
In your notation, you denote by di the dimension of the i-th subspace (as numbered above) and (for some reason) you should write the dimensions as a 5-tuple (d1,d2,…,d5). So let us compute.
1) Diagonal matrix is such that all non-diagonal entries of the matrix are zero. This means that the diagonal entries may be arbitrary. We have 29 rows and 29 columns, thus 29 diagonal entries which are independent of each other. Hence the dimension is
d1=29 .
2) Upper triangular matrix is such that all the entries on the diagonal and below the diagonal are zero. Again, the entries above the diagonal are arbitrary real numbers. So we have to compute how many positions we have above the diagonal for a 29×29 matrix. The number of such positions is given by the following sum
28+27+26+⋯+3+2+1
and corresponds to the number of free slots in the first row, second row, third row, and so on. Check for example arbitrary 4×4 upper triangular matrix:
⎛⎝⎜⎜⎜0000a000bd00cef0⎞⎠⎟⎟⎟
and the number of the above diagonal entries is 3+2+1=6. For the case of 29×29 matrix, the corresponding sum may be computed by the formula
n2(a1+an) ,
where in our case n=28,a1=28,an=1. Thus we have
d2=28+27+26+⋯+3+2+1=242⋅(28+1)=406 .
3) Trace-zero matrix is the matrix having zero trace (does make sense right? :P) In other words, if we sum up the diagonal entries (this corresponds to computing the trace) we get zero. This implies two things: i) all the entries but one on the diagonal are arbitrary ; ii) the of the diagonal entries may be arbitrary. The later should be clear. The former is true because of the following: denote by a1,a2,…,an the diagonal entries of a given matrix. The trace-zero condition means
a1+a2+⋯+an=0 .
Moving arbitrary ai (for simplicity the last one, an) on the right side of the equation gives
a1+a2+⋯+an−1=−an .
This implies that the first n−1 diagonal entries may be chosen arbitrarily, forcing the last one to have the value −(a1+a2+⋯+an−1). Altogether we have that all but one entries of a trace-zero matrix are arbitrary, meaning the dimension of the coresponding subspace is
d3=n2−1=292−1=840 ,
where the n2 in the formula arises from the fact that each n×n matrix consists of n2 entries.
4) Symmetric matrix is such that At=A where At is the transpose of A. In other words, A is symmetric if the entries below and above the diagonal are the same on the corresponding places, i.e. the following condition holds (see the wiki article about the transpose of a matrix)
aij=aji .
So the entries on the diagonal are arbitrary and the entries above the diagonal completely determine the entries below the diagonal. We have already computed the number of diagonal entries as well as the above diagonal entries (see cases 1) and 2) ). Thus the dimension of the subspace of 29×29 symmetric matrices is
d4=29+406=435 ,
where the first summand corresponds to the diagonal and the second summand to the above diagonal entries.
5) Skew-symmetric matrix (see wiki article on skew-symmetric matrices) is such that At=−A which amounts to those matrices which has entries aij satisfying the condition
aij=−aji .
The above condition implies
i) aii=−aii which means that the diagonal entries must be zero;
ii) the elements above the diagonal completely determines the elements below the diagonal.
This means that the number of skew-symmetric matrices is the same as the number of upper triangular matrices (since only the elements above the diagonal may be arbitrary), that is
d5=406 .
Wrapping it all up, the 5-tupple you are searching for is
(29,406,840,435,406) .