Question

Find the dimensions of the following
The space of n x n matrices all of whose elements are 0 except possibly the diagonal elements.
The space of n x n upper triangular matrices
The space of n x n symmetric matrices
The space of n x n diagonal matrices

Answer 1

1) The space of n x n matrices all of whose elements are 0 except possibly the diagonal elements has dimensions n. This is because there can be at most n non-zero elements in the diagonal, and any combination of values for these elements represents a distinct matrix in the space.

2) The space of n x n upper triangular matrices has dimensions n(n+1)/2. This is because an upper triangular matrix has all elements below the main diagonal equal to 0. The number of elements in the main diagonal is n, and the number of elements above the main diagonal is given by the sum of the first n natural numbers, which is n(n+1)/2.

3) The space of n x n symmetric matrices also has dimensions n(n+1)/2. This is because a symmetric matrix is equal to its transpose, so there are constraints on the elements above the main diagonal. Similar to the upper triangular matrix, the number of elements in the main diagonal is n, and the number of elements above the main diagonal is given by the sum of the first n natural numbers, which is n(n+1)/2.

4) The space of n x n diagonal matrices has dimensions n. This is because a diagonal matrix only has non-zero elements along the main diagonal, and there are n such elements. The rest of the elements in the matrix are all 0.