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
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.
To find the dimensions of the given spaces, we need to determine the number of independent parameters or variables required to define each space.
1. The space of n x n matrices all of whose elements are 0 except possibly the diagonal elements:
In such matrices, all elements except the diagonal elements are fixed at zero. For an n x n matrix, there are n diagonal elements that can vary independently. Therefore, the dimension of this space is n.
2. The space of n x n upper triangular matrices:
An upper triangular matrix is a square matrix in which all the elements below the main diagonal (i.e., elements with row index greater than column index) are zero. For an upper triangular matrix, the main diagonal and all the elements above it are the independent variables. The number of elements in the main diagonal is n, and each element above it requires one additional independent variable. Hence, the dimension of this space is n + (n-1) + (n-2) + ... + 2 + 1, which simplifies to n(n+1)/2.
3. The space of n x n symmetric matrices:
A symmetric matrix is a square matrix that is equal to its transpose (i.e., the element at the i-th row and j-th column is equal to the element at the j-th row and i-th column). In a symmetric matrix, the diagonal elements and the upper triangular portion (including the main diagonal) are the independent variables. Similar to the upper triangular matrices, the diagonal has n elements, and each element above it requires one additional independent variable. Hence, the dimension of this space is n + (n-1) + (n-2) + ... + 2 + 1, which simplifies to n(n+1)/2.
4. The space of n x n diagonal matrices:
A diagonal matrix is a square matrix in which all off-diagonal elements are zero. In this case, all the diagonal elements (i.e., elements with the same row and column indices) are the independent variables. Therefore, the dimension of this space is n.
In summary:
1. Dimensions of space of n x n matrices with 0 except possibly diagonal elements: n
2. Dimensions of space of n x n upper triangular matrices: n(n+1)/2
3. Dimensions of space of n x n symmetric matrices: n(n+1)/2
4. Dimensions of space of n x n diagonal matrices: n