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Concept
Diagonal matrix
Summary
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.
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