Table of contents |
2 Eigenvectors, eigenvalues, determinant 3 Uses |
The operations of matrix addition and matrix multiplication are especially simple for diagonal matrices. Write diag(a1,...,an) for a diagonal matrix whose diagonal entries starting in the upper left corner are a1,...,an. Then, for addition, we have
Matrix operations
and for matrix multiplication,
Multiplying the matrix A from the left with diag(a1,...,an) amounts to multiplying the i-th row of A by ai for all i; multiplying the matrix A from the right with diag(a1,...,an) amounts to multiplying the i-th column of A by ai for all i.
The eigenvalues of diag(a1,...,an) are a1,...,an. The unit vectors e1,...,en form a basis of eigenvectors. The determinant of diag(a1,...,an) is the product a1...an.
Diagonal matrices occur in many areas of linear algebra. Because of the simple description of the matrix operation and eigenvalues/eigenvectors given above, it is always desirable to represent a given matrix or linear map by a diagonal matrix.
In fact, a given n-by-n matrix is similar to a diagonal matrix if and only if it has n linearly independent eigenvectors. These matrices are called diagonalizable.
Over the field of real or complex numbers, more is true: every normal matrix is unitarily similar to a diagonal matrix (the spectral theorem), and every matrix is unitarily equivalent to a diagonal matrix with nonnegative entries (the singular value decomposition).Eigenvectors, eigenvalues, determinant
Uses