Definition: Symmetric matrix. A symmetric matrix is equal to its transpose. An anti-symmetric matrix is equal to minus its transpose. All symmetric matrices are square matrices.
- For a symmetric matrix,
, etc.
Definition: Anti-symmetric:
All anti-symmetric matrices are also square matrices.
- For all anti-symmetric matrices, the diagonal elements are zero.
- For any matrix
, you can write it as the sum of symmetric and anti-symmetric matrices
Definition: Triangular matrix. An upper triangular matrix is a square matrix with nonzero entries on or above the main diagonal. A lower triangular matrix is a square matrix with nonzero entries only on or below the main diagonal.
- Again, all of the triangular matrices are square matrices
- For some
, Any element is below the diagonal, then their row index must be larger than their column index. is the row index, and is the column index
- So, a matrix
is upper triangular if , where . - A matrix
is lower triangular is , where .
Definition: Diagonal matrix. A diagonal matrix is a square matrix with nonzero entries (if any) only on the main diagonal (top left to bottom right). Similar to the identity matrices
, multiplied by some constant
- Another way of defining these matrices is
, , if . - Diagonal matrices are symmetric, anti-symmetric, and triangular
- If
is symmetric, and is invertible, does that force to be symmetric? - If
is anti-symmetric, same question - If a diagonal matrix
is invertible, is necessarily diagonal?
- If