# Diagonal matrix

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Matrix whose only nonzero elements are on its main diagonal

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In [linear algebra](/source/Linear_algebra), a **diagonal matrix** is a [matrix](/source/Matrix_(mathematics)) in which the entries outside the [main diagonal](/source/Main_diagonal) are all zero; the term usually refers to [square matrices](/source/Square_matrices). Elements of the main diagonal can either be zero or nonzero. An example of a 2×2 diagonal matrix is [ 3 0 0 2 ] {\displaystyle \left[{\begin{smallmatrix}3&0\\0&2\end{smallmatrix}}\right]} , while an example of a 3×3 diagonal matrix is [ 6 0 0 0 5 0 0 0 4 ] {\displaystyle \left[{\begin{smallmatrix}6&0&0\\0&5&0\\0&0&4\end{smallmatrix}}\right]} . An [identity matrix](/source/Identity_matrix) of any size, or any multiple of it is a diagonal matrix called a [*scalar matrix*](#Scalar_matrix), for example, [ 0.5 0 0 0.5 ] {\displaystyle \left[{\begin{smallmatrix}0.5&0\\0&0.5\end{smallmatrix}}\right]} . In [geometry](/source/Geometry), a diagonal matrix may be used as a *[scaling matrix](/source/Scaling_matrix)*, since matrix multiplication with it results in changing scale (size) and possibly also [shape](/source/Shape); only a scalar matrix results in uniform change in scale.

## Definition

As stated above, a diagonal matrix is a matrix in which all off-diagonal entries are zero. That is, the matrix **D** = (*d**i*,*j*) with n columns and n rows is diagonal if ∀ i , j ∈ { 1 , 2 , … , n } , i ≠ j ⟹ d i , j = 0. {\displaystyle \forall i,j\in \{1,2,\ldots ,n\},i\neq j\implies d_{i,j}=0.}

However, the main diagonal entries are unrestricted.

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: [ 1 0 0 0 4 0 0 0 − 3 0 0 0 ] or [ 1 0 0 0 0 0 4 0 0 0 0 0 − 3 0 0 ] {\displaystyle {\begin{bmatrix}1&0&0\\0&4&0\\0&0&-3\\0&0&0\\\end{bmatrix}}\quad {\text{or}}\quad {\begin{bmatrix}1&0&0&0&0\\0&4&0&0&0\\0&0&-3&0&0\end{bmatrix}}}

More often, however, *diagonal matrix* refers to square matrices, which can be specified explicitly as a **square diagonal matrix**. A square diagonal matrix is a [symmetric matrix](/source/Symmetric_matrix), so this can also be called a **symmetric diagonal matrix**.

The following matrix is square diagonal matrix: [ 1 0 0 0 4 0 0 0 − 2 ] {\displaystyle {\begin{bmatrix}1&0&0\\0&4&0\\0&0&-2\end{bmatrix}}}

If the entries are [real numbers](/source/Real_numbers) or [complex numbers](/source/Complex_numbers), then it is a [normal matrix](/source/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".

## Vector-to-matrix diag operator

A diagonal matrix **D** can be constructed from a vector a = [ a 1 … a n ] T {\displaystyle \mathbf {a} ={\begin{bmatrix}a_{1}&\dots &a_{n}\end{bmatrix}}^{\textsf {T}}} using the diag {\displaystyle \operatorname {diag} } operator: D = diag ⁡ ( a 1 , … , a n ) . {\displaystyle \mathbf {D} =\operatorname {diag} (a_{1},\dots ,a_{n}).}

This may be written more compactly as D = diag ⁡ ( a ) {\displaystyle \mathbf {D} =\operatorname {diag} (\mathbf {a} )} .

The same operator is also used to represent [block diagonal matrices](/source/Block_matrix#Block_diagonal_matrices) as A = diag ⁡ ( A 1 , … , A n ) {\displaystyle \mathbf {A} =\operatorname {diag} (\mathbf {A} _{1},\dots ,\mathbf {A} _{n})} where each argument **A***i* is a matrix.

The diag operator may be written as diag ⁡ ( a ) = ( a 1 T ) ∘ I , {\displaystyle \operatorname {diag} (\mathbf {a} )=\left(\mathbf {a} \mathbf {1} ^{\textsf {T}}\right)\circ \mathbf {I} ,} where ∘ {\displaystyle \circ } represents the [Hadamard product](/source/Hadamard_product_(matrices)), and **1** is a constant vector with elements 1.

## Matrix-to-vector diag operator

The inverse matrix-to-vector diag operator is sometimes denoted by the identically named diag ⁡ ( D ) = [ a 1 … a n ] T , {\displaystyle \operatorname {diag} (\mathbf {D} )={\begin{bmatrix}a_{1}&\dots &a_{n}\end{bmatrix}}^{\textsf {T}},} where the argument is now a matrix, and the result is a vector of its diagonal entries.

The following property holds: diag ⁡ ( A B ) = ∑ j ( A ∘ B T ) i j = ( A ∘ B T ) 1 . {\displaystyle \operatorname {diag} (\mathbf {A} \mathbf {B} )=\sum _{j}\left(\mathbf {A} \circ \mathbf {B} ^{\textsf {T}}\right)_{ij}=\left(\mathbf {A} \circ \mathbf {B} ^{\textsf {T}}\right)\mathbf {1} .}

## Scalar matrix

A diagonal matrix with equal diagonal entries is a **scalar matrix**; that is, a scalar multiple λ of the [identity matrix](/source/Identity_matrix) **I**. Its effect on a [vector](/source/Vector_(mathematics_and_physics)) is [scalar multiplication](/source/Scalar_multiplication) by λ. For example, a 3×3 scalar matrix has the form: [ λ 0 0 0 λ 0 0 0 λ ] ≡ λ I 3 {\displaystyle {\begin{bmatrix}\lambda &0&0\\0&\lambda &0\\0&0&\lambda \end{bmatrix}}\equiv \lambda {\boldsymbol {I}}_{3}}

The scalar matrices are the [center](/source/Center_of_an_algebra) of the algebra of matrices: that is, they are precisely the matrices that [commute](/source/Commute_(mathematics)) with all other square matrices of the same size.[a] By contrast, over a [field](/source/Field_(mathematics)) (like the real numbers), a diagonal matrix with all diagonal elements distinct only commutes with diagonal matrices (its [centralizer](/source/Centralizer) is the set of diagonal matrices). That is because if a diagonal matrix D = diag ⁡ ( a 1 , … , a n ) {\displaystyle \mathbf {D} =\operatorname {diag} (a_{1},\dots ,a_{n})} has a i ≠ a j , {\displaystyle a_{i}\neq a_{j},} then given a matrix **M** with m i j ≠ 0 , {\displaystyle m_{ij}\neq 0,} the (*i*, *j*) term of the products are: ( D M ) i j = a i m i j {\displaystyle (\mathbf {DM} )_{ij}=a_{i}m_{ij}} and ( M D ) i j = m i j a j , {\displaystyle (\mathbf {MD} )_{ij}=m_{ij}a_{j},} and a j m i j ≠ m i j a i {\displaystyle a_{j}m_{ij}\neq m_{ij}a_{i}} (since one can divide by mij), so they do not commute unless the off-diagonal terms are zero.[b] Diagonal matrices where the diagonal entries are not all equal or all distinct have centralizers intermediate between the whole space and only diagonal matrices.[1]

For an abstract vector space V (rather than the concrete vector space Kn), the analog of scalar matrices are **scalar transformations**. This is true more generally for a [module](/source/Module_(ring_theory)) M over a [ring](/source/Ring_(algebra)) R, with the [endomorphism algebra](/source/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 R → End ⁡ ( M ) , {\displaystyle R\to \operatorname {End} (M),} (from a scalar λ to its corresponding scalar transformation, multiplication by λ) exhibiting End(*M*) as a R-[algebra](/source/Algebra_(ring_theory)). For vector spaces, the scalar transforms are exactly the [center](/source/Center_of_a_ring) of the endomorphism algebra, and, similarly, scalar invertible transforms are the center of the [general linear group](/source/General_linear_group) GL(*V*). The former is more generally true [free modules](/source/Free_module) M ≅ R n , {\displaystyle M\cong R^{n},} 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 D = diag ⁡ ( a 1 , … , a n ) {\displaystyle \mathbf {D} =\operatorname {diag} (a_{1},\dots ,a_{n})} and a vector v = [ x 1 ⋯ x n ] T {\displaystyle \mathbf {v} ={\begin{bmatrix}x_{1}&\dotsm &x_{n}\end{bmatrix}}^{\textsf {T}}} , the product is: D v = diag ⁡ ( a 1 , … , a n ) [ x 1 ⋮ x n ] = [ a 1 ⋱ a n ] [ x 1 ⋮ x n ] = [ a 1 x 1 ⋮ a n x n ] . {\displaystyle \mathbf {D} \mathbf {v} =\operatorname {diag} (a_{1},\dots ,a_{n}){\begin{bmatrix}x_{1}\\\vdots \\x_{n}\end{bmatrix}}={\begin{bmatrix}a_{1}\\&\ddots \\&&a_{n}\end{bmatrix}}{\begin{bmatrix}x_{1}\\\vdots \\x_{n}\end{bmatrix}}={\begin{bmatrix}a_{1}x_{1}\\\vdots \\a_{n}x_{n}\end{bmatrix}}.}

This can be expressed more compactly by using a vector instead of a diagonal matrix, d = [ a 1 ⋯ a n ] T {\displaystyle \mathbf {d} ={\begin{bmatrix}a_{1}&\dotsm &a_{n}\end{bmatrix}}^{\textsf {T}}} , and taking the [Hadamard product](/source/Hadamard_product_(matrices)) of the vectors (entrywise product), denoted d ∘ v {\displaystyle \mathbf {d} \circ \mathbf {v} } :

D v = d ∘ v = [ a 1 ⋮ a n ] ∘ [ x 1 ⋮ x n ] = [ a 1 x 1 ⋮ a n x n ] . {\displaystyle \mathbf {D} \mathbf {v} =\mathbf {d} \circ \mathbf {v} ={\begin{bmatrix}a_{1}\\\vdots \\a_{n}\end{bmatrix}}\circ {\begin{bmatrix}x_{1}\\\vdots \\x_{n}\end{bmatrix}}={\begin{bmatrix}a_{1}x_{1}\\\vdots \\a_{n}x_{n}\end{bmatrix}}.}

This is mathematically equivalent, but avoids storing all the zero terms of this [sparse matrix](/source/Sparse_matrix). This product is thus used in [machine learning](/source/Machine_learning), such as computing products of derivatives in [backpropagation](/source/Backpropagation) or multiplying IDF weights in [TF-IDF](/source/TF-IDF),[2] since some [BLAS](/source/BLAS) frameworks, which multiply matrices efficiently, do not include Hadamard product capability directly.[3]

## Matrix operations

The operations of matrix addition and [matrix multiplication](/source/Matrix_multiplication) are especially simple for diagonal matrices. Write diag(*a*1, ..., *an*) for a diagonal matrix whose diagonal entries starting in the upper left corner are *a*1, ..., *an*. Then, for [addition](/source/Matrix_addition), we have

diag ⁡ ( a 1 , … , a n ) + diag ⁡ ( b 1 , … , b n ) = diag ⁡ ( a 1 + b 1 , … , a n + b n ) {\displaystyle \operatorname {diag} (a_{1},\,\ldots ,\,a_{n})+\operatorname {diag} (b_{1},\,\ldots ,\,b_{n})=\operatorname {diag} (a_{1}+b_{1},\,\ldots ,\,a_{n}+b_{n})}

and for [matrix multiplication](/source/Matrix_multiplication),

diag ⁡ ( a 1 , … , a n ) diag ⁡ ( b 1 , … , b n ) = diag ⁡ ( a 1 b 1 , … , a n b n ) . {\displaystyle \operatorname {diag} (a_{1},\,\ldots ,\,a_{n})\operatorname {diag} (b_{1},\,\ldots ,\,b_{n})=\operatorname {diag} (a_{1}b_{1},\,\ldots ,\,a_{n}b_{n}).}

The diagonal matrix diag(*a*1, ..., *an*) is [invertible](/source/Invertible_matrix) [if and only if](/source/If_and_only_if) the entries *a*1, ..., *an* are all nonzero. In this case, we have

diag ⁡ ( a 1 , … , a n ) − 1 = diag ⁡ ( a 1 − 1 , … , a n − 1 ) . {\displaystyle \operatorname {diag} (a_{1},\,\ldots ,\,a_{n})^{-1}=\operatorname {diag} (a_{1}^{-1},\,\ldots ,\,a_{n}^{-1}).}

In particular, the diagonal matrices form a [subring](/source/Subring) of the ring of all n-by-n matrices.

Multiplying an n-by-n matrix **A** from the *left* with diag(*a*1, ..., *an*) amounts to multiplying the i-th *row* of **A** by ai for all i; multiplying the matrix **A** from the *right* with diag(*a*1, ..., *an*) amounts to multiplying the i-th *column* of **A** by ai for all i.

## Operator matrix in eigenbasis

Main articles: [Transformation matrix § Finding the matrix of a transformation](/source/Transformation_matrix#Finding_the_matrix_of_a_transformation), and [Eigenvalues and eigenvectors](/source/Eigenvalues_and_eigenvectors)

As explained in [determining coefficients of operator matrix](/source/Transformation_matrix#Finding_the_matrix_of_a_transformation), there is a special basis, **e**1, ..., **e***n*, for which the matrix **A** takes the diagonal form. Hence, in the defining equation A e j = ∑ i a i , j e i {\textstyle \mathbf {Ae} _{j}=\sum _{i}a_{i,j}\mathbf {e} _{i}} , all coefficients ai, j with *i* ≠ *j* are zero, leaving only one term per sum. The surviving diagonal elements, ai, j, are known as **eigenvalues** and designated with λi in the equation, which reduces to A e i = λ i e i . {\displaystyle \mathbf {Ae} _{i}=\lambda _{i}\mathbf {e} _{i}.} The resulting equation is known as **eigenvalue equation**[4] and used to derive the [characteristic polynomial](/source/Characteristic_polynomial) and, further, [eigenvalues and eigenvectors](/source/Eigenvalues_and_eigenvectors).

In other words, the [eigenvalues](/source/Eigenvalue) of diag(*λ*1, ..., *λ**n*) are *λ*1, ..., *λ**n* with associated [eigenvectors](/source/Eigenvectors) of **e**1, ..., **e***n*.

## Properties

- The [determinant](/source/Determinant) of diag(*a*1, ..., *a**n*) is the product *a*1⋯*a**n*.

- The [adjugate](/source/Adjugate) of a diagonal matrix is again diagonal.

- Where all matrices are square, - A matrix is diagonal if and only if it is triangular and [normal](/source/Normal_matrix). - A matrix is diagonal if and only if it is both [upper-](/source/Triangular_matrix) and [lower-triangular](/source/Triangular_matrix). - A diagonal matrix is [symmetric](/source/Symmetric_matrix).

- The [identity matrix](/source/Identity_matrix) **I***n* and [zero matrix](/source/Zero_matrix) are diagonal.

- A 1×1 matrix is always diagonal.

- The square of a 2×2 matrix with zero [trace](/source/Trace_(linear_algebra)) is always diagonal.

- The inverse of a diagonal matrix is obtained by taking the reciprocals of the elements on the diagonal

## 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](/source/Linear_operator) by a diagonal matrix.

In fact, a given n-by-n matrix **A** is [similar](/source/Similar_matrix) 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](/source/Linearly_independent) eigenvectors. Such matrices are said to be [diagonalizable](/source/Diagonalizable_matrix).

Over the [field](/source/Field_(mathematics)) of [real](/source/Real_number) or [complex](/source/Complex_number) numbers, more is true. The [spectral theorem](/source/Spectral_theorem) says that every [normal matrix](/source/Normal_matrix) is [unitarily similar](/source/Matrix_similarity) to a diagonal matrix (if **AA**∗ = **A**∗**A** then there exists a [unitary matrix](/source/Unitary_matrix) **U** such that **UAU**∗ is diagonal). Furthermore, the [singular value decomposition](/source/Singular_value_decomposition) implies that for any matrix **A**, there exist unitary matrices **U** and **V** such that **U**∗**AV** is diagonal with positive entries.

## Operator theory

In [operator theory](/source/Operator_theory), particularly the study of [PDEs](/source/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](/source/Separable_partial_differential_equation). Therefore, a key technique to understanding operators is a change of coordinates—in the language of operators, an [integral transform](/source/Integral_transform)—which changes the basis to an [eigenbasis](/source/Eigenbasis) of [eigenfunctions](/source/Eigenfunction): which makes the equation separable. An important example of this is the [Fourier transform](/source/Fourier_transform), which diagonalizes constant coefficient differentiation operators (or more generally translation invariant operators), such as the Laplacian operator, say, in the [heat equation](/source/Heat_equation).

Especially easy are [multiplication operators](/source/Multiplication_operator), 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.

## See also

- [Anti-diagonal matrix](/source/Anti-diagonal_matrix)

- [Banded matrix](/source/Banded_matrix)

- [Bidiagonal matrix](/source/Bidiagonal_matrix)

- [Diagonally dominant matrix](/source/Diagonally_dominant_matrix)

- [Diagonalizable matrix](/source/Diagonalizable_matrix)

- [Jordan normal form](/source/Jordan_normal_form)

- [Multiplication operator](/source/Multiplication_operator)

- [Tridiagonal matrix](/source/Tridiagonal_matrix)

- [Toeplitz matrix](/source/Toeplitz_matrix)

- [Toral Lie algebra](/source/Toral_Lie_algebra)

- [Circulant matrix](/source/Circulant_matrix)

## Notes

1. **[^](#cite_ref-1)** Proof: given the [elementary matrix](/source/Elementary_matrix) e i j {\displaystyle e_{ij}} , M e i j {\displaystyle Me_{ij}} is the matrix with only the *i*-th row of *M* and e i j M {\displaystyle e_{ij}M} is the square matrix with only the *M* *j*-th column, so the non-diagonal entries must be zero, and the *i*th diagonal entry much equal the *j*th diagonal entry.

1. **[^](#cite_ref-2)** Over more general rings, this does not hold, because one cannot always divide.

## References

1. **[^](#cite_ref-3)** ["Do Diagonal Matrices Always Commute?"](https://math.stackexchange.com/q/1697991). Stack Exchange. March 15, 2016. Retrieved August 4, 2018.

1. **[^](#cite_ref-4)** Sahami, Mehran (2009-06-15). [*Text Mining: Classification, Clustering, and Applications*](https://books.google.com/books?id=BnvYaYhMl-MC&pg=PA14). CRC Press. p. 14. [ISBN](/source/ISBN_(identifier)) [9781420059458](https://en.wikipedia.org/wiki/Special:BookSources/9781420059458).

1. **[^](#cite_ref-5)** ["Element-wise vector-vector multiplication in BLAS?"](https://stackoverflow.com/questions/7621520/element-wise-vector-vector-multiplication-in-blas). *stackoverflow.com*. 2011-10-01. Retrieved 2020-08-30.

1. **[^](#cite_ref-6)** Nearing, James (2010). ["Chapter 7.9: Eigenvalues and Eigenvectors"](http://www.physics.miami.edu/~nearing/mathmethods/operators.pdf) (PDF). [*Mathematical Tools for Physics*](http://www.physics.miami.edu/nearing/mathmethods). Dover Publications. [ISBN](/source/ISBN_(identifier)) [978-0486482125](https://en.wikipedia.org/wiki/Special:BookSources/978-0486482125). Retrieved January 1, 2012.

## Sources

- [Horn, Roger Alan](/source/Roger_Horn); [Johnson, Charles Royal](/source/Charles_Royal_Johnson) (1985), *Matrix Analysis*, [Cambridge University Press](/source/Cambridge_University_Press), [ISBN](/source/ISBN_(identifier)) [978-0-521-38632-6](https://en.wikipedia.org/wiki/Special:BookSources/978-0-521-38632-6)

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Adapted from the Wikipedia article [Diagonal matrix](https://en.wikipedia.org/wiki/Diagonal_matrix) by Wikipedia contributors ([contributor history](https://en.wikipedia.org/wiki/Diagonal_matrix?action=history)). Available under [Creative Commons Attribution-ShareAlike 4.0 International](https://creativecommons.org/licenses/by-sa/4.0/). Changes may have been made.