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. An example of a 2×2 diagonal matrix is formula_1, while an example of a 3×3 diagonal matrix isformula_2. 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.
Definition.
As stated above, a diagonal matrix is a matrix in which all off-diagonal entries are zero. That is, the matrix 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 "d""i","i" being zero. For example:
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".
Scalar matrix.
A diagonal matrix with equal diagonal entries is a scalar matrix; that is, a scalar multiple "λ" 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. By contrast, over a field (like the real numbers), a diagonal matrix with all diagonal elements distinct only commutes with diagonal matrices (its centralizer is the set of diagonal matrices). That is because if a diagonal matrix formula_8 has formula_9 then given a matrix formula_10 with formula_11 the formula_12 term of the products are: formula_13 and formula_14 and formula_15 (since one can divide by formula_16), so they do not commute unless the off-diagonal terms are zero. Diagonal matrices where the diagonal entries are not all equal or all distinct have centralizers intermediate between the whole space and only diagonal matrices.
For an abstract vector space "V" (rather than the concrete vector space formula_17), the analog of scalar matrices are scalar transformations. This is true more generally for a module "M" over a ring "R", with the endomorphism algebra End("M") (algebra of linear operators on "M") replacing the algebra of matrices. Formally, scalar multiplication is a linear map, inducing a map formula_18 (from a scalar "λ" to its corresponding scalar transformation, multiplication by "λ") exhibiting End("M") as a "R"-algebra. For vector spaces, 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"). The former is more generally true free modules formula_19, for which the endomorphism algebra is isomorphic to a matrix algebra.
Vector operations.
Multiplying a vector by a diagonal matrix multiplies each of the terms by the corresponding diagonal entry. Given a diagonal matrix formula_8 and a vector formula_21, the product is:
This can be expressed more compactly by using a vector instead of a diagonal matrix, formula_23, and taking the Hadamard product of the vectors (entrywise product), denoted formula_24:
This is mathematically equivalent, but avoids storing all the zero terms of this sparse matrix. This product is thus used in machine learning, such as computing products of derivatives in backpropagation or multiplying IDF weights in TF-IDF, since some BLAS frameworks, which multiply matrices efficiently, do not include Hadamard product capability directly.
Matrix operations.
The operations of matrix addition and matrix multiplication are especially simple for diagonal matrices. Write for a diagonal matrix whose diagonal entries starting in the upper left corner are "a"1, ..., "a""n". Then, for addition, we have
and for matrix multiplication,
The diagonal matrix is invertible if and only if the entries "a"1, ..., "a""n" are all non-zero. In this case, we have
In particular, the diagonal matrices form a subring of the ring of all "n"-by-"n" matrices.
Multiplying an "n"-by-"n" matrix "A" from the "left" with amounts to multiplying the "i"th "row" of "A" by "a""i" for all "i"; multiplying the matrix "A" from the "right" with amounts to multiplying the "i"th "column" of "A" by "a""i" for all "i".
Operator matrix in eigenbasis.
As explained in determining coefficients of operator matrix, there is a special basis, e1, …, e"n", for which the matrix formula_26 takes the diagonal form. Hence, in the defining equation formula_27, all coefficients formula_28 with "i" ≠ "j" are zero, leaving only one term per sum. The surviving diagonal elements, formula_29, are known as eigenvalues and designated with formula_30 in the equation, which reduces to formula_31. The resulting equation is known as eigenvalue equation and used to derive the characteristic polynomial and, further, eigenvalues and eigenvectors.
In other words, the eigenvalues of are "λ"1, …, "λ""n" with associated eigenvectors of e1, …, e"n".
Applications.
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 typically 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 "X"−1"AX" 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.
Operator theory.
In operator theory, particularly the study of PDEs, operators are particularly easy to understand and PDEs easy to solve if the operator is diagonal with respect to the basis with which one is working; this corresponds to a separable partial differential equation. Therefore, a key technique to understanding operators is a change of coordinates—in the language of operators, an integral transform—which changes the basis to an eigenbasis of eigenfunctions: which makes the equation separable. An important example of this is the Fourier transform, which diagonalizes constant coefficient differentiation operators (or more generally translation invariant operators), such as the Laplacian operator, say, in the heat equation.
Especially easy are multiplication operators, which are defined as multiplication by (the values of) a fixed function–the values of the function at each point correspond to the diagonal entries of a matrix.