Identities for matrix commutators

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Problem 15.19 from Беклемишева Л.А., Беклемишев Д.В., Петрович А.Ю., Чубаров И.А. "Сборник задач по аналитической геометрии и линейной алгебре" [ББП+23]More problems in Subsection 15.1 "Main operations with matrices: multiplication by a number, addition and multiplication of matrices (15.1-15.24)"

(1)(2)(3)(4)

Verify the validity of the identity (see problem 15. 18):

(1)

$\left[A, B\right] = - \left[B, A\right]$

(2)

$\left[A, A\right] = O$

(3)

$\left[A, E\right] = \left[E, A\right] = O$

(4)

$\left[A, (B + C)\right] = \left[A, B\right] + \left[A, C\right]$

Problem 17.23 from Кострикин А.И. "Сборник задач по алгебре" [Кос15]More problems in Section 17 "Operations on matrices"

(a)(b)

Prove that for any square matrices $A, B, C$ the following equalities hold:

(a)

$[A, B C]=[A, B] C+B[A, C]$ ;

(b)

$[[A, B], C]+[[B, C], A]+[[C, A], B]=0$ .

Problem II.2.4.13 from Кострикин А.И. "Линейная алгебра" [Кос00]More problems in Section II.2.4 "Jordan normal form"

Prove that $\left[[X, Y]^{2}, Z\right]=0$ for any three matrices $X, Y, Z \in M_{2}(\mathfrak {K})$ . Here $[X, Y]=X Y-Y X$ is the commutator of the matrices $X, Y$ .

Problem 4.1.16 from Steeb W., Hardy Y. "Problems and Solutions in Introductory and Advanced Matrix Calculus" [StHa17]More problems in Section 4.1 "Chapter Problems"

(i)(ii)

(i)

Let $A$ , $B$ be $n \times n$ matrices. Show that

\operatorname {tr}((A+B)(A-B)) \equiv \operatorname {tr}(A^{2})-\operatorname {tr}(B^{2}).

(ii)

Let $A$ , $B$ , $C$ be $n \times n$ matrices. Show that $\operatorname {tr}([A,B]C) = \operatorname {tr}(A[B,C])$ .

Problem 7.1.5 from Steeb W., Hardy Y. "Problems and Solutions in Introductory and Advanced Matrix Calculus" [StHa17]More problems in Section 7.1 "Chapter Problems"

Let $A$ , $B$ , $H$ be $n \times n$ matrices over $\mathbb {C}$ such that $[A, H] = 0_n$ , $[B, H] = 0_n$ . Find $[[A, B], H]$ .

Problem 7.1.9 from Steeb W., Hardy Y. "Problems and Solutions in Introductory and Advanced Matrix Calculus" [StHa17]More problems in Section 7.1 "Chapter Problems"

(i)(ii)(iii)

(i)

Let $A$ , $B$ be $n \times n$ matrices over $\mathbb {C}$ . Let $S$ be an invertible $n \times n$ matrix over $\mathbb {C}$ with $\widetilde{A} = S^{-1}AS$ , $\widetilde{B} = S^{-1}BS$ . Show that $[\widetilde{A}, \widetilde{B}] = S^{-1}[A, B]S$ .

(ii)

Let $A$ , $B$ be $n \times n$ matrices over $\mathbb {C}$ . Assume that $[A, B] = 0_n$ . Let $U$ be a unitary matrix. Calculate $[U^{*}AU, U^{*}BU]$ .

(iii)

Let $T$ be an invertible matrix. Show that $T^{-1}AT \equiv A + T^{-1}[A, T]$ .

Problem 7.1.24 from Steeb W., Hardy Y. "Problems and Solutions in Introductory and Advanced Matrix Calculus" [StHa17]More problems in Section 7.1 "Chapter Problems"

Let $A$ , $B$ be $n \times n$ matrices. Given the expression

A^{2}B + AB^{2} + B^{2}A + BA^{2} - 2ABA - 2BAB.

Write the expression in a more compact form using commutators.

Problem 7.1.26 from Steeb W., Hardy Y. "Problems and Solutions in Introductory and Advanced Matrix Calculus" [StHa17]More problems in Section 7.1 "Chapter Problems"

Let $A_1$ , $A_2$ , $A_3$ be $n \times n$ matrices over $\mathbb {C}$ . The ternary commutator $[A_1, A_2, A_3]$ (also called the ternutator) is defined as

[A_1, A_2, A_3] := \sum _{\pi \in S_3} \operatorname {sign}\left(\pi \right) A_{\pi (1)}A_{\pi (2)}A_{\pi (3)} \equiv A_1A_2A_3 + A_2A_3A_1 + A_3A_1A_2 - A_1A_3A_2 - A_2A_1A_3 - A_3A_2A_1.

Let $n = 2$ and consider the Pauli spin matrices $\sigma_1$ , $\sigma_2$ , $\sigma_3$ . Calculate the ternutator $[\sigma_1, \sigma_2, \sigma_3]$ .

Problem 7.1.29 from Steeb W., Hardy Y. "Problems and Solutions in Introductory and Advanced Matrix Calculus" [StHa17]More problems in Section 7.1 "Chapter Problems"

Let $A$ , $B$ be $n \times n$ matrices over $\mathbb {C}$ . Assume that $B$ is invertible. Show that

[A, B^{-1}] \equiv -B^{-1}[A, B]B^{-1}.

Problem 7.2.21 from Steeb W., Hardy Y. "Problems and Solutions in Introductory and Advanced Matrix Calculus" [StHa17]More problems in Section 7.2 "Supplementary Problems"

Let $n \geq 1$ and $A$ , $B$ be $n \times n$ matrices over $\mathbb {C}$ . Show that

[A, B^{n}] = \sum _{j=0}^{n-1} A^{j}[A, B]B^{n-1-j}.

Problem 7.2.25 from Steeb W., Hardy Y. "Problems and Solutions in Introductory and Advanced Matrix Calculus" [StHa17]More problems in Section 7.2 "Supplementary Problems"

Let $A$ , $B$ be $n \times n$ matrices and $T$ a (fixed) invertible $n \times n$ matrix. We define the bracket

[A, B]_T := ATB - BTA.

Let

T = \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}, \qquad X = \begin{pmatrix} 0 & 1 \\ 0 & 0 \end{pmatrix}, \qquad Y = \begin{pmatrix} 0 & 0 \\ 1 & 0 \end{pmatrix}, \qquad H = \begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix}.

Find $[X, Y]_T$ , $[X, H]_T$ , $[Y, H]_T$ .

Problem 7.2.30 from Steeb W., Hardy Y. "Problems and Solutions in Introductory and Advanced Matrix Calculus" [StHa17]More problems in Section 7.2 "Supplementary Problems"

Let $A$ , $B$ , $C$ be $n \times n$ matrices. Show that

\operatorname {tr}([A, B]C) \equiv \operatorname {tr}(A[B, C])

applying cyclic invariance.

Problem 3.20 from Steeb W., Tanski I., Hardy Y. "Problems and Solutions for Groups, Lie Groups, Lie Algebras with Applications" [STH12]More problems in Chapter 3 "Lie Algebras"

(i)(ii)

(i)

Let $A$ be an associative algebra over a field $\mathbb {F}$ . We define the commutator

[a,b] := ab - ba

for all $a, b \in A$ . Show that with this definition of the commutator we have a Lie algebra.

(ii)

Give an example.

Problem 3.80 from Steeb W., Tanski I., Hardy Y. "Problems and Solutions for Groups, Lie Groups, Lie Algebras with Applications" [STH12]More problems in Chapter 3 "Lie Algebras"

(i)(ii)

(i)

Let $U, V$ be $n \times n$ unitary matrices such that the commutator $[U,V] = W$ is again a unitary matrix, i.e. $W^{*} = W^{-1}$ . Show that

R + R^{-1} = I_{n}

where $R = U^{-1}V^{-1}UV$ is the group commutator of $U^{-1}$ and $V^{-1}$ .

(ii)

Give an example for such matrices $U, V$ for the case $n=2$ .

Problem 3.87 from Steeb W., Tanski I., Hardy Y. "Problems and Solutions for Groups, Lie Groups, Lie Algebras with Applications" [STH12]More problems in Chapter 3 "Lie Algebras"

(i)(ii)

Let $A_{1}, A_{2}, A_{3}$ be $n \times n$ matrices over $\mathbb {C}$ . The ternary commutator $[A_{1},A_{2},A_{3}]$ (also called the ternutator) is defined as

\begin{align} [A_{1},A_{2},A_{3}] & := \sum _{\pi \in S_{3}} \operatorname {sgn}(\pi ) A_{\pi (1)}A_{\pi (2)}A_{\pi (3)} \\ & \equiv A_{1}A_{2}A_{3} + A_{2}A_{3}A_{1} + A_{3}A_{1}A_{2} - A_{1}A_{3}A_{2} - A_{2}A_{1}A_{3} - A_{3}A_{2}A_{1} \, . \end{align}

(i)

Let $n=2$ and consider the Pauli spin matrices $\sigma_{x}, \sigma_{y}, \sigma_{z}$ . Calculate the ternutator

[\sigma _{x}, \sigma _{y}, \sigma _{z}] \, .

(ii)

Calculate

A_{1}\otimes A_{2}\otimes A_{3} + A_{2}\otimes A_{3}\otimes A_{1} + A_{3}\otimes A_{1}\otimes A_{2} - A_{1}\otimes A_{3}\otimes A_{2} - A_{2}\otimes A_{1}\otimes A_{3} - A_{3}\otimes A_{2}\otimes A_{1} \, .

References

[ББП+23] Беклемишева Л.А., Беклемишев Д.В., Петрович А.Ю., Чубаров И.А. Сборник задач по аналитической геометрии и линейной алгебре. Лань, 2023.

[Кос15] Кострикин А.И. Сборник задач по алгебре. МЦНМО, 2015.

[Кос00] Кострикин А.И. Линейная алгебра. ФИЗМАТЛИТ, 2000.

[StHa17] Steeb W., Hardy Y. Problems and Solutions in Introductory and Advanced Matrix Calculus. 2nd ed., World Scientific Publishing, 2017.

[STH12] Steeb W., Tanski I., Hardy Y. Problems and Solutions for Groups, Lie Groups, Lie Algebras with Applications. World Scientific Publishing, 2012.