Diagonal matrix

In linear algebra, a diagonal matrix is a matrix in which the entries outside the main diagonal are all zero; the term usually refers to square matrices. Elements of the main diagonal can either be zero or nonzero. An example of a 2×2 diagonal matrix is , while an example of a 3×3 diagonal matrix is. An identity matrix of any size, or any multiple of it (a scalar matrix), is a diagonal matrix. A diagonal matrix is sometimes called a scaling matrix, since matrix multiplication with it results in changing scale (size). Its determinant is the product of its diagonal values. As stated above, a diagonal matrix is a matrix in which all off-diagonal entries are zero. That is, the matrix D = (di,j) with n columns and n rows is diagonal if However, the main diagonal entries are unrestricted. The term diagonal matrix may sometimes refer to a , which is an m-by-n matrix with all the entries not of the form di,i being zero. For example: or More often, however, diagonal matrix refers to square matrices, which can be specified explicitly as a . A square diagonal matrix is a symmetric matrix, so this can also be called a . The following matrix is square 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 diagonal matrices, and refer to them simply as "diagonal matrices". A diagonal matrix can be constructed from a vector using the operator: This may be written more compactly as . The same operator is also used to represent block diagonal matrices as where each argument is a matrix. The operator may be written as: where represents the Hadamard product and is a constant vector with elements 1. The inverse matrix-to-vector operator is sometimes denoted by the identically named where the argument is now a matrix and the result is a vector of its diagonal entries. The following property holds: A diagonal matrix with equal diagonal entries is a scalar matrix; that is, a scalar multiple λ of the identity matrix I.