# Triangular matrix

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Special kind of square matrix

Not to be confused with a [triangular array](/source/Triangular_array), a related concept.

For the rings, see [triangular matrix ring](/source/Triangular_matrix_ring).

"Triangularization" and "triangularizable" redirect here. For the geometric processes, see [Triangulation](/source/Triangulation) and [Triangulation (topology)](/source/Triangulation_(topology)).

In mathematics, a **triangular matrix** is a special kind of [square matrix](/source/Square_matrix). A square matrix is called **lower triangular** if all the entries *above* the [main diagonal](/source/Main_diagonal) are zero. Similarly, a square matrix is called **upper triangular** if all the entries *below* the main diagonal are zero.

Because matrix equations with triangular matrices are easier to solve, they are very important in [numerical analysis](/source/Numerical_analysis). By the [LU decomposition](/source/LU_decomposition) algorithm, an [invertible matrix](/source/Invertible_matrix) may be written as the [product](/source/Matrix_multiplication) of a lower triangular matrix *L* and an upper triangular matrix *U* [if and only if](/source/If_and_only_if) all its leading principal [minors](/source/Minor_(linear_algebra)) are non-zero.

## Description

A matrix of the form

- L = [ ℓ 1 , 1 0 ℓ 2 , 1 ℓ 2 , 2 ℓ 3 , 1 ℓ 3 , 2 ⋱ ⋮ ⋮ ⋱ ⋱ ℓ n , 1 ℓ n , 2 … ℓ n , n − 1 ℓ n , n ] {\displaystyle L={\begin{bmatrix}\ell _{1,1}&&&&0\\\ell _{2,1}&\ell _{2,2}&&&\\\ell _{3,1}&\ell _{3,2}&\ddots &&\\\vdots &\vdots &\ddots &\ddots &\\\ell _{n,1}&\ell _{n,2}&\ldots &\ell _{n,n-1}&\ell _{n,n}\end{bmatrix}}}

is called a **lower triangular matrix** or **left triangular matrix**, and analogously a matrix of the form

- U = [ u 1 , 1 u 1 , 2 u 1 , 3 … u 1 , n u 2 , 2 u 2 , 3 … u 2 , n ⋱ ⋱ ⋮ ⋱ u n − 1 , n 0 u n , n ] {\displaystyle U={\begin{bmatrix}u_{1,1}&u_{1,2}&u_{1,3}&\ldots &u_{1,n}\\&u_{2,2}&u_{2,3}&\ldots &u_{2,n}\\&&\ddots &\ddots &\vdots \\&&&\ddots &u_{n-1,n}\\0&&&&u_{n,n}\end{bmatrix}}}

is called an **upper triangular matrix** or **right triangular matrix**. A lower or left triangular matrix is commonly denoted with the variable *L*, and an upper or right triangular matrix is commonly denoted with the variable *U* or *R*.

A matrix that is both upper and lower triangular is [diagonal](/source/Diagonal_matrix). Matrices that are [similar](/source/Similar_(linear_algebra)) to triangular matrices are called **triangularisable**.

A non-square (or sometimes any) matrix with zeros above (below) the diagonal is called a lower (upper) trapezoidal matrix. The non-zero entries form the shape of a [trapezoid](/source/Trapezoid).

### Examples

The matrix

- [ 1 0 0 2 96 0 4 9 69 ] {\displaystyle {\begin{bmatrix}1&0&0\\2&96&0\\4&9&69\end{bmatrix}}}

is the lower triangular for the non symmetric matrix:

- [ 1 5 8 2 96 9 4 9 69 ] {\displaystyle {\begin{bmatrix}1&5&8\\2&96&9\\4&9&69\end{bmatrix}}}

and

- [ 1 4 1 0 6 9 0 0 1 ] {\displaystyle {\begin{bmatrix}1&4&1\\0&6&9\\0&0&1\end{bmatrix}}}

is the upper triangular for the non symmetric matrix:

- [ 1 4 1 99 6 9 40 88 1 ] {\displaystyle {\begin{bmatrix}1&4&1\\99&6&9\\40&88&1\end{bmatrix}}}

## Forward and back substitution

A matrix equation in the form L x = b {\displaystyle L\mathbf {x} =\mathbf {b} } or U x = b {\displaystyle U\mathbf {x} =\mathbf {b} } is very easy to solve by an iterative process called **forward substitution** for lower triangular matrices and analogously **back substitution** for upper triangular matrices. The process is so called because for lower triangular matrices, one first computes x 1 {\displaystyle x_{1}} , then substitutes that *forward* into the *next* equation to solve for x 2 {\displaystyle x_{2}} , and repeats through to x n {\displaystyle x_{n}} . In an upper triangular matrix, one works *backwards,* first computing x n {\displaystyle x_{n}} , then substituting that *back* into the *previous* equation to solve for x n − 1 {\displaystyle x_{n-1}} , and repeating through x 1 {\displaystyle x_{1}} .

Notice that this does not require inverting the matrix.

### Forward substitution

The matrix equation *L***x** = **b** can be written as a system of linear equations

- ℓ 1 , 1 x 1 = b 1 ℓ 2 , 1 x 1 + ℓ 2 , 2 x 2 = b 2 ⋮ ⋮ ⋱ ⋮ ℓ m , 1 x 1 + ℓ m , 2 x 2 + ⋯ + ℓ m , m x m = b m {\displaystyle {\begin{matrix}\ell _{1,1}x_{1}&&&&&&&=&b_{1}\\\ell _{2,1}x_{1}&+&\ell _{2,2}x_{2}&&&&&=&b_{2}\\\vdots &&\vdots &&\ddots &&&&\vdots \\\ell _{m,1}x_{1}&+&\ell _{m,2}x_{2}&+&\dotsb &+&\ell _{m,m}x_{m}&=&b_{m}\\\end{matrix}}}

Observe that the first equation ( ℓ 1 , 1 x 1 = b 1 {\displaystyle \ell _{1,1}x_{1}=b_{1}} ) only involves x 1 {\displaystyle x_{1}} , and thus one can solve for x 1 {\displaystyle x_{1}} directly. The second equation only involves x 1 {\displaystyle x_{1}} and x 2 {\displaystyle x_{2}} , and thus can be solved once one substitutes in the already solved value for x 1 {\displaystyle x_{1}} . Continuing in this way, the k {\displaystyle k} -th equation only involves x 1 , … , x k {\displaystyle x_{1},\dots ,x_{k}} , and one can solve for x k {\displaystyle x_{k}} using the previously solved values for x 1 , … , x k − 1 {\displaystyle x_{1},\dots ,x_{k-1}} . The resulting formulas are:

- x 1 = b 1 ℓ 1 , 1 , x 2 = b 2 − ℓ 2 , 1 x 1 ℓ 2 , 2 , ⋮ x m = b m − ∑ i = 1 m − 1 ℓ m , i x i ℓ m , m . {\displaystyle {\begin{aligned}x_{1}&={\frac {b_{1}}{\ell _{1,1}}},\\x_{2}&={\frac {b_{2}-\ell _{2,1}x_{1}}{\ell _{2,2}}},\\&\ \ \vdots \\x_{m}&={\frac {b_{m}-\sum _{i=1}^{m-1}\ell _{m,i}x_{i}}{\ell _{m,m}}}.\end{aligned}}}

A matrix equation with an upper triangular matrix *U* can be solved in an analogous way, only working backwards.

### Applications

Forward substitution is used in financial [bootstrapping](/source/Bootstrapping_(finance)) to construct a [yield curve](/source/Yield_curve).

## Properties

The [transpose](/source/Transpose) of an upper triangular matrix is a lower triangular matrix and vice versa.

A matrix which is both symmetric and triangular is diagonal. In a similar vein, a matrix which is both [normal](/source/Normal_matrix) (meaning *A***A* = *AA**, where *A** is the [conjugate transpose](/source/Conjugate_transpose)) and triangular is also diagonal. This can be seen by looking at the diagonal entries of *A***A* and *AA**.

The [determinant](/source/Determinant) and [permanent](/source/Permanent_(mathematics)) of a triangular matrix equal the product of the diagonal entries, as can be checked by direct computation.

In fact more is true: the [eigenvalues](/source/Eigenvalue) of a triangular matrix are exactly its diagonal entries. Moreover, each eigenvalue occurs exactly *k* times on the diagonal, where *k* is its [algebraic multiplicity](/source/Algebraic_multiplicity), that is, its [multiplicity as a root](/source/Multiplicity_of_a_root_of_a_polynomial) of the [characteristic polynomial](/source/Characteristic_polynomial) p A ( x ) = det ( x I − A ) {\displaystyle p_{A}(x)=\det(xI-A)} of *A*. In other words, the characteristic polynomial of a triangular *n*×*n* matrix *A* is exactly

- p A ( x ) = ( x − a 11 ) ( x − a 22 ) ⋯ ( x − a n n ) {\displaystyle p_{A}(x)=(x-a_{11})(x-a_{22})\cdots (x-a_{nn})} ,

that is, the unique degree *n* polynomial whose roots are the diagonal entries of *A* (with multiplicities). To see this, observe that x I − A {\displaystyle xI-A} is also triangular and hence its determinant det ( x I − A ) {\displaystyle \det(xI-A)} is the product of its diagonal entries ( x − a 11 ) ( x − a 22 ) ⋯ ( x − a n n ) {\displaystyle (x-a_{11})(x-a_{22})\cdots (x-a_{nn})} .[1]

## Special forms

### Unitriangular matrix

If the entries on the [main diagonal](/source/Main_diagonal) of a (lower or upper) triangular matrix are all 1, the matrix is called (lower or upper) **unitriangular**.

Other names used for these matrices are **unit** (lower or upper) **triangular**, or very rarely **normed** (lower or upper) **triangular**. However, a *unit* triangular matrix is not the same as **the** *[unit matrix](/source/Identity_matrix)*, and a *normed* triangular matrix has nothing to do with the notion of [matrix norm](/source/Matrix_norm).

All finite unitriangular matrices are [unipotent](/source/Unipotent).

### Strictly triangular matrix

If all of the entries on the main diagonal of a (lower or upper) triangular matrix are also 0, the matrix is called **strictly** (lower or upper) **triangular**.

All finite strictly triangular matrices are [nilpotent](/source/Nilpotent_matrix) of index at most *n* as a consequence of the [Cayley–Hamilton theorem](/source/Cayley%E2%80%93Hamilton_theorem).

### Atomic triangular matrix

Main article: [Frobenius matrix](/source/Frobenius_matrix)

An **atomic** (lower or upper) **triangular matrix** is a special form of unitriangular matrix, where all of the [off-diagonal elements](/source/Off-diagonal_element) are zero, except for the entries in a single column. Such a matrix is also called a **Frobenius matrix**, a **Gauss matrix**, or a **Gauss transformation matrix**.

### Block triangular matrix

Main article: [Block matrix](/source/Block_matrix)

A block triangular matrix is a [block matrix](/source/Block_matrix) (partitioned matrix) that is a triangular matrix.

#### Upper block triangular

A matrix A {\displaystyle A} is **upper block triangular** if

- A = [ A 11 A 12 ⋯ A 1 k 0 A 22 ⋯ A 2 k ⋮ ⋮ ⋱ ⋮ 0 0 ⋯ A k k ] {\displaystyle A={\begin{bmatrix}A_{11}&A_{12}&\cdots &A_{1k}\\0&A_{22}&\cdots &A_{2k}\\\vdots &\vdots &\ddots &\vdots \\0&0&\cdots &A_{kk}\end{bmatrix}}} ,

where A i j ∈ F n i × n j {\displaystyle A_{ij}\in \mathbb {F} ^{n_{i}\times n_{j}}} for all i , j = 1 , … , k {\displaystyle i,j=1,\ldots ,k} .[2]

#### Lower block triangular

A matrix A {\displaystyle A} is **lower block triangular** if

- A = [ A 11 0 ⋯ 0 A 21 A 22 ⋯ 0 ⋮ ⋮ ⋱ ⋮ A k 1 A k 2 ⋯ A k k ] {\displaystyle A={\begin{bmatrix}A_{11}&0&\cdots &0\\A_{21}&A_{22}&\cdots &0\\\vdots &\vdots &\ddots &\vdots \\A_{k1}&A_{k2}&\cdots &A_{kk}\end{bmatrix}}} ,

where A i j ∈ F n i × n j {\displaystyle A_{ij}\in \mathbb {F} ^{n_{i}\times n_{j}}} for all i , j = 1 , … , k {\displaystyle i,j=1,\ldots ,k} .[2]

## Triangularisability

A matrix that is [similar](/source/Similar_matrix) to a triangular matrix is referred to as **triangularizable**. Abstractly, this is equivalent to stabilizing a [flag](/source/Flag_(linear_algebra)): upper triangular matrices are precisely those that preserve the [standard flag](/source/Standard_flag), which is given by the standard ordered basis ( e 1 , … , e n ) {\displaystyle (e_{1},\ldots ,e_{n})} and the resulting flag 0 < ⟨ e 1 ⟩ < ⟨ e 1 , e 2 ⟩ < ⋯ < ⟨ e 1 , … , e n ⟩ = K n . {\displaystyle 0<\left\langle e_{1}\right\rangle <\left\langle e_{1},e_{2}\right\rangle <\cdots <\left\langle e_{1},\ldots ,e_{n}\right\rangle =K^{n}.} All flags are conjugate (as the [general linear group](/source/General_linear_group) acts transitively on bases), so any matrix that stabilises a flag is similar to one that stabilizes the standard flag.

Any complex square matrix is triangularizable.[1] In fact, a matrix *A* over a [field](/source/Field_(mathematics)) containing all of the eigenvalues of *A* (for example, any matrix over an [algebraically closed field](/source/Algebraically_closed_field)) is similar to a triangular matrix. This can be proven by using induction on the fact that *A* has an eigenvector, by taking the quotient space by the eigenvector and inducting to show that *A* stabilizes a flag, and is thus triangularizable with respect to a basis for that flag.

A more precise statement is given by the [Jordan normal form](/source/Jordan_normal_form) theorem, which states that in this situation, *A* is similar to an upper triangular matrix of a very particular form. The simpler triangularization result is often sufficient however, and in any case used in proving the Jordan normal form theorem.[1][3]

In the case of complex matrices, it is possible to say more about triangularization, namely, that any square matrix *A* has a [Schur decomposition](/source/Schur_decomposition). This means that *A* is unitarily equivalent (i.e. similar, using a [unitary matrix](/source/Unitary_matrix) as change of basis) to an upper triangular matrix; this follows by taking an Hermitian basis for the flag.

### Simultaneous triangularisability

See also: [Simultaneously diagonalizable](/source/Simultaneously_diagonalizable)

A set of matrices A 1 , … , A k {\displaystyle A_{1},\ldots ,A_{k}} are said to be **simultaneously triangularisable** if there is a basis under which they are all upper triangular; equivalently, if they are upper triangularizable by a single similarity matrix *P.* Such a set of matrices is more easily understood by considering the algebra of matrices it generates, namely all polynomials in the A i , {\displaystyle A_{i},} denoted K [ A 1 , … , A k ] . {\displaystyle K[A_{1},\ldots ,A_{k}].} Simultaneous triangularizability means that this algebra is conjugate into the Lie subalgebra of upper triangular matrices, and is equivalent to this algebra being a Lie subalgebra of a [Borel subalgebra](/source/Borel_subalgebra).

The basic result is that (over an algebraically closed field), the [commuting matrices](/source/Commuting_matrices) A , B {\displaystyle A,B} or more generally A 1 , … , A k {\displaystyle A_{1},\ldots ,A_{k}} are simultaneously triangularizable. This can be proven by first showing that commuting matrices have a common eigenvector, and then inducting on dimension as before. This was proven by Frobenius, starting in 1878 for a commuting pair, as discussed at [commuting matrices](/source/Commuting_matrices). As for a single matrix, over the complex numbers these can be triangularized by unitary matrices.

The fact that commuting matrices have a common eigenvector can be interpreted as a result of [Hilbert's Nullstellensatz](/source/Hilbert's_Nullstellensatz): commuting matrices form a commutative algebra K [ A 1 , … , A k ] {\displaystyle K[A_{1},\ldots ,A_{k}]} over K [ x 1 , … , x k ] {\displaystyle K[x_{1},\ldots ,x_{k}]} which can be interpreted as a variety in *k*-dimensional affine space, and the existence of a (common) eigenvalue (and hence a common eigenvector) corresponds to this variety having a point (being non-empty), which is the content of the (weak) Nullstellensatz.[*[citation needed](https://en.wikipedia.org/wiki/Wikipedia:Citation_needed)*] In algebraic terms, these operators correspond to an [algebra representation](/source/Algebra_representation) of the polynomial algebra in *k* variables.

This is generalized by [Lie's theorem](/source/Lie's_theorem), which shows that any representation of a [solvable Lie algebra](/source/Solvable_Lie_algebra) is simultaneously upper triangularizable, the case of commuting matrices being the [abelian Lie algebra](/source/Abelian_Lie_algebra) case, abelian being a fortiori solvable.

More generally and precisely, a set of matrices A 1 , … , A k {\displaystyle A_{1},\ldots ,A_{k}} is simultaneously triangularisable if and only if the matrix p ( A 1 , … , A k ) [ A i , A j ] {\displaystyle p(A_{1},\ldots ,A_{k})[A_{i},A_{j}]} is [nilpotent](/source/Nilpotent) for all polynomials *p* in *k* *non*-commuting variables, where [ A i , A j ] {\displaystyle [A_{i},A_{j}]} is the [commutator](/source/Commutator); for commuting A i {\displaystyle A_{i}} the commutator vanishes so this holds. This was proven by Drazin, Dungey, and Gruenberg in 1951;[4] a brief proof is given by Prasolov in 1994.[5] One direction is clear: if the matrices are simultaneously triangularisable, then [ A i , A j ] {\displaystyle [A_{i},A_{j}]} is *strictly* upper triangularizable (hence nilpotent), which is preserved by multiplication by any A k {\displaystyle A_{k}} or combination thereof – it will still have 0s on the diagonal in the triangularizing basis.

## Algebras of triangular matrices

[Binary](/source/Logical_matrix) lower unitriangular [Toeplitz](/source/Toeplitz_matrix) matrices, multiplied using [**F**2](/source/Finite_field) operations. They form the [Cayley table](/source/Cayley_table) of [Z4](/source/Cyclic_group) and correspond to [powers of the 4-bit Gray code permutation](https://en.wikiversity.org/wiki/Gray_code_permutation_powers#4_bit).

Upper triangularity is preserved by many operations:

- The sum of two upper triangular matrices is upper triangular.

- The product of two upper triangular matrices is upper triangular.

- The [inverse](/source/Inverse_matrix) of an upper triangular matrix, if it exists, is upper triangular.

- The product of an upper triangular matrix and a scalar is upper triangular.

Together these facts mean that the upper triangular matrices form a [subalgebra](/source/Subalgebra) of the [associative algebra](/source/Associative_algebra) of square matrices for a given size. Additionally, this also shows that the upper triangular matrices can be viewed as a Lie subalgebra of the [Lie algebra](/source/Lie_algebra) of square matrices of a fixed size, where the [Lie bracket](/source/Lie_bracket) [*a*, *b*] given by the [commutator](/source/Commutator#Ring_theory) *ab − ba*. The Lie algebra of all upper triangular matrices is a [solvable Lie algebra](/source/Solvable_Lie_algebra). It is often referred to as a [Borel subalgebra](/source/Borel_subalgebra) of the Lie algebra of all square matrices.

All these results hold if *upper triangular* is replaced by *lower triangular* throughout; in particular the lower triangular matrices also form a Lie algebra. However, operations mixing upper and lower triangular matrices do not in general produce triangular matrices. For instance, the sum of an upper and a lower triangular matrix can be any matrix; the product of a lower triangular with an upper triangular matrix is not necessarily triangular either.

The set of unitriangular matrices forms a [Lie group](/source/Lie_group).

The set of strictly upper (or lower) triangular matrices forms a [nilpotent Lie algebra](/source/Nilpotent_Lie_algebra), denoted n . {\displaystyle {\mathfrak {n}}.} This algebra is the [derived Lie algebra](/source/Derived_Lie_algebra) of b {\displaystyle {\mathfrak {b}}} , the Lie algebra of all upper triangular matrices; in symbols, n = [ b , b ] . {\displaystyle {\mathfrak {n}}=[{\mathfrak {b}},{\mathfrak {b}}].} In addition, n {\displaystyle {\mathfrak {n}}} is the Lie algebra of the Lie group of unitriangular matrices.

In fact, by [Engel's theorem](/source/Engel's_theorem), any finite-dimensional nilpotent Lie algebra is conjugate to a subalgebra of the strictly upper triangular matrices, that is to say, a finite-dimensional nilpotent Lie algebra is simultaneously strictly upper triangularizable.

Algebras of upper triangular matrices have a natural generalization in [functional analysis](/source/Functional_analysis) which yields [nest algebras](/source/Nest_algebra) on [Hilbert spaces](/source/Hilbert_space).

See also: [Affine group](/source/Affine_group)

### Borel subgroups and Borel subalgebras

Main articles: [Borel subgroup](/source/Borel_subgroup) and [Borel subalgebra](/source/Borel_subalgebra)

The set of invertible triangular matrices of a given kind (lower or upper) forms a [group](/source/Group_(mathematics)), indeed a [Lie group](/source/Lie_group), which is a subgroup of the [general linear group](/source/General_linear_group) of all invertible matrices. A triangular matrix is invertible precisely when its diagonal entries are invertible (non-zero).

Over the real numbers, this group is disconnected, having 2 n {\displaystyle 2^{n}} components accordingly as each diagonal entry is positive or negative. The identity component is invertible triangular matrices with positive entries on the diagonal, and the group of all invertible triangular matrices is a [semidirect product](/source/Semidirect_product) of this group and the group of [diagonal matrices](/source/Diagonal_matrix) with ± 1 {\displaystyle \pm 1} on the diagonal, corresponding to the components.

The [Lie algebra](/source/Lie_algebra) of the Lie group of invertible upper triangular matrices is the set of all upper triangular matrices, not necessarily invertible, and is a [solvable Lie algebra](/source/Solvable_Lie_algebra). These are, respectively, the standard [Borel subgroup](/source/Borel_subgroup) *B* of the Lie group GL*n* and the standard [Borel subalgebra](/source/Borel_subalgebra) b {\displaystyle {\mathfrak {b}}} of the Lie algebra gl*n*.

The upper triangular matrices are precisely those that stabilize the [standard flag](/source/Flag_(linear_algebra)). The invertible ones among them form a subgroup of the general linear group, whose conjugate subgroups are those defined as the stabilizer of some (other) complete flag. These subgroups are [Borel subgroups](/source/Borel_subgroup). The group of invertible lower triangular matrices is such a subgroup, since it is the stabilizer of the standard flag associated to the standard basis in reverse order.

The stabilizer of a partial flag obtained by forgetting some parts of the standard flag can be described as a set of block upper triangular matrices (but its elements are *not* all triangular matrices). The conjugates of such a group are the subgroups defined as the stabilizer of some partial flag. These subgroups are called parabolic subgroups.

### Examples

The group of 2×2 upper unitriangular matrices is [isomorphic](/source/Isomorphic) to the [additive group](/source/Abelian_group) of the field of scalars; in the case of complex numbers it corresponds to a group formed of parabolic [Möbius transformations](/source/M%C3%B6bius_transformation); the 3×3 upper unitriangular matrices form the [Heisenberg group](/source/Heisenberg_group).

## See also

- [Gaussian elimination](/source/Gaussian_elimination)

- [QR decomposition](/source/QR_decomposition)

- [Cholesky decomposition](/source/Cholesky_decomposition)

- [Hessenberg matrix](/source/Hessenberg_matrix)

- [Tridiagonal matrix](/source/Tridiagonal_matrix)

- [Invariant subspace](/source/Invariant_subspace)

## References

1. ^ [***a***](#cite_ref-axler_1-0) [***b***](#cite_ref-axler_1-1) [***c***](#cite_ref-axler_1-2) Axler, Sheldon Jay (1997). *Linear Algebra Done Right* (2nd ed.). New York: Springer. pp. 86–87, 169. [ISBN](/source/ISBN_(identifier)) [0-387-22595-1](https://en.wikipedia.org/wiki/Special:BookSources/0-387-22595-1). [OCLC](/source/OCLC_(identifier)) [54850562](https://search.worldcat.org/oclc/54850562).

1. ^ [***a***](#cite_ref-bernstein2009_2-0) [***b***](#cite_ref-bernstein2009_2-1) Bernstein, Dennis S. (2009). *Matrix mathematics: theory, facts, and formulas* (2 ed.). Princeton, NJ: Princeton University Press. p. 168. [ISBN](/source/ISBN_(identifier)) [978-0-691-14039-1](https://en.wikipedia.org/wiki/Special:BookSources/978-0-691-14039-1).

1. **[^](#cite_ref-herstein_3-0)** Herstein, I. N. (1975). *Topics in Algebra* (2nd ed.). New York: Wiley. pp. 285–290. [ISBN](/source/ISBN_(identifier)) [0-471-01090-1](https://en.wikipedia.org/wiki/Special:BookSources/0-471-01090-1). [OCLC](/source/OCLC_(identifier)) [3307396](https://search.worldcat.org/oclc/3307396).

1. **[^](#cite_ref-4)** Drazin, M. P.; Dungey, J. W.; Gruenberg, K. W. (1951). ["Some Theorems on Commutative Matrices"](http://jlms.oxfordjournals.org/cgi/pdf_extract/s1-26/3/221). *Journal of the London Mathematical Society*. **26** (3): 221–228. [doi](/source/Doi_(identifier)):[10.1112/jlms/s1-26.3.221](https://doi.org/10.1112%2Fjlms%2Fs1-26.3.221).

1. **[^](#cite_ref-5)** Prasolov, V. V. (1994). *Problems and Theorems in Linear Algebra*. Simeon Ivanov. Providence, R.I.: American Mathematical Society. pp. 178–179. [ISBN](/source/ISBN_(identifier)) [9780821802366](https://en.wikipedia.org/wiki/Special:BookSources/9780821802366). [OCLC](/source/OCLC_(identifier)) [30076024](https://search.worldcat.org/oclc/30076024).

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