Diagonal matrices

In linear algebra, a diagonal matrix is a matrix in which the entries outside the main diagonal are all zero.

As stated above, the off-diagonal entries are zero. That is, the matrix D = (d_i,j) with n columns and n rows is diagonal if

However, the main diagonal entries need not be zero.

The term diagonal matrix may sometimes refer to a rectangular diagonal matrix, which is an m-by-n matrix with all the entries not of the form d_i,i being zero. For example:

The following matrix is a symmetric diagonal matrix:

If the entries are real numbers or complex numbers, then it is a normal matrix as well.

In the remainder of this article we will consider only square matrices.

A square 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. For example, a 3×3 scalar matrix has the form:

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.

For an abstract vector space V (rather than the concrete vector space ${\displaystyle K^{n}}$), or more generally a module M over a ring R, with the endomorphism algebra End(M) (algebra of linear operators on M) replacing the algebra of matrices, the analog of scalar matrices are scalar transformations. Formally, scalar multiplication is a linear map, inducing a map ${\displaystyle R\to \operatorname {End} (M),}$ (send a scalar λ to the corresponding scalar transformation, multiplication by λ) exhibiting End(M) as a R-algebra. For vector spaces, or more generally free modules ${\displaystyle M\cong R^{n}}$, for which the endomorphism algebra is isomorphic to a matrix algebra, the scalar transforms are exactly the center of the endomorphism algebra, and similarly invertible transforms are the center of the general linear group GL(V), where they are denoted by Z(V), follow the usual notation for the center.

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