For example, the following matrix is diagonal:
The term diagonal matrix may sometimes refer to a rectangular diagonal matrix, which is an m-by-n matrix with only the entries of the form di,i possibly non-zero; for example,
However, in the remainder of this article we will consider only square matrices.
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.
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
and for matrix multiplication,
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.
The eigenvalues of diag(a1, ..., an) are a1, ..., an with associated eigenvectors of e1, ..., en, where the vector ei is all zeros except a one in the ith row. The determinant of diag(a1, ..., an) is the product a1...an.
The adjugate of a diagonal matrix is again diagonal.
A square matrix is diagonal if and only if it is triangular and normal.
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.