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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 entries themselves may or may not be zero. Thus, the matrix D = (di,j) 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 In is diagonal.

Table of contents
1 Matrix operations
2 Eigenvectors, eigenvalues, determinant
3 Uses

Matrix operations

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

diag(a1,...,an) + diag(b1,...,bn) = diag(a1+b1,...,an+bn)

and for matrix multiplication,

diag(a1,...,an) · diag(b1,...,bn) = diag(a1b1,...,anbn).

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

diag(a1,...,an)-1 = diag(a1-1,...,an-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,...,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.

Eigenvectors, eigenvalues, determinant

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


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).