coconut62 said:
A diagonal matrix is a square matrix in which all the elements outside of the main diagonal are equal to zero. The main diagonal refers to the elements that run from the top left to the bottom right of the matrix.
In a diagonal matrix, the eigenvalues are arranged along the main diagonal from top left to bottom right. This means that the first eigenvalue corresponds to the top left element, the second eigenvalue corresponds to the second element on the diagonal, and so on.
Yes, all diagonal matrices are square matrices because they have an equal number of rows and columns. This is a necessary condition for a matrix to be considered diagonal.
No, a diagonal matrix, by definition, has all zero elements outside of the main diagonal. If a matrix has non-zero elements outside of the main diagonal, it is not considered to be a diagonal matrix.
A diagonal matrix is significant in linear algebra because it represents a transformation that stretches or compresses the axes of a vector space. It is also useful in solving systems of linear equations and in calculating the determinant and inverse of a matrix.
