Diagonal matrix

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

<math>d_{i,j} = 0 \mbox{ if } i \ne j </math>

Example:

<math>\begin{bmatrix}

1 & 0 \\ 0 & 4 \end{bmatrix}</math>

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.

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.

Eigenvectors, eigenvalues, determinant

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.

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 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).