In linear algebra, a diagonal matrix is a square matrix in which only the entries in the main diagonal are non-zero. The diagonal entries themselves may or may not be zero. Thus, the matrix D = (d_i,j) with n columns and n rows is diagonal if:

For example, the following matrix is diagonal:

Any diagonal matrix is also a symmetric matrix, a triangular matrix, and (if the entries come from the field R or C) also a normal matrix. The identity matrix I_n is diagonal.

A diagonal matrix with all its main diagonal entries equal is a scalar matrix, that is, a scalar multiple λI of the identity matrix I. Its effect on a vector is scalar multiplication by λ. The scalar matrices are the center of the algebra of matrices: that is, they are precisely the matrices that commute with all other square matrices of the same size.

Matrix operations

The operations of matrix addition and matrix multiplication are especially simple for diagonal matrices. Write diag(a₁,...,a_n) for a diagonal matrix whose diagonal entries starting in the upper left corner are a₁,...,a_n. Then, for addition, we have

diag(a₁,...,a_n) + diag(b₁,...,b_n) = diag(a₁+b₁,...,a_n+b_n)

and for matrix multiplication,

diag(a₁,...,a_n) · diag(b₁,...,b_n) = diag(a₁b₁,...,a_nb_n).

The diagonal matrix diag(a₁,...,a_n) is invertible if and only if the entries a₁,...,a_n are all non-zero. In this case, we have

diag(a₁,...,a_n)^-1 = diag(a₁^-1,...,a_n^-1).

In particular, the diagonal matrices form a subring of the ring of all n-by-n matrices.

Multiplying the matrix A from the left with diag(a₁,...,a_n) amounts to multiplying the i-th row of A by a_i for all i; multiplying the matrix A from the right with diag(a₁,...,a_n) amounts to multiplying the i-th column of A by a_i for all i.

Other properties

The eigenvalues of diag(a₁, ..., a_n) are a₁, ..., a_n. The unit vectors e₁, ..., e_n form a basis of eigenvectors. The determinant of diag(a₁, ..., a_n) is the product a₁...a_n.

The adjugate of a diagonal matrix is again diagonal.

A square matrix is diagonal if and only if it is triangular and normal.

Uses

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 A is similar to a diagonal matrix (meaning that there is a matrix X such that XAX^-1 is diagonal) if and only if it has n linearly independent eigenvectors. Such matrices are said to be diagonalizable.

Over the field of real or complex numbers, more is true. The spectral theorem says that every normal matrix is unitarily similar to a diagonal matrix (if AA^* = A^*A then there exists a unitary matrix U such that UAU^* is diagonal). Furthermore, the singular value decomposition implies that for any matrix A, there exist unitary matrices U and V such that UAV^* is diagonal with positive entries.

External links

References

• Roger A. Horn and Charles R. Johnson, Matrix Analysis, Cambridge University Press, 1985. ISBN 0-521-30586-1 (hardback), ISBN 0-521-38632-2 (paperback).