diagonaalimatriisille

A diagonaalimatriisi is a square matrix where all the entries outside of the main diagonal are zero. The main diagonal consists of the elements from the top-left corner to the bottom-right corner of the matrix. A matrix D is a diagonaalimatriisi if D_ij = 0 for all i ≠ j, where D_ij represents the element in the i-th row and j-th column. The elements on the main diagonal, D_ii, can be any scalar value, including zero.

Diagonaalimatriisit have several useful properties in linear algebra. For instance, the determinant of a diagonaalimatriisi is simply the product of its diagonal entries. The inverse of a non-singular diagonaalimatriisi is also a diagonaalimatriisi, with the reciprocal of each diagonal entry. Matrix multiplication involving diagonaalimatriisit is simplified; when multiplying two diagonaalimatriisit, the result is another diagonaalimatriisi whose diagonal entries are the products of the corresponding diagonal entries of the original matrices.

Eigenvalues of a diagonaalimatriisi are its diagonal entries, and the corresponding eigenvectors are the standard basis vectors. This makes diagonaalimatriisit particularly important in the study of eigenvalues and eigenvectors, and in the diagonalization of matrices. They also appear in various applications, such as in the representation of linear transformations that scale axes independently.