Triangular form and invariant flags

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Problem 22.16 from Harville D.A. "Matrix Algebra: Exercises and Solutions" [Har01]More problems in Chapter 22 "Linear Transformations"

(a)(b)

Let $T$ represent a linear transformation from an $n$ -dimensional linear space $\mathcal{V}$ into $\mathcal{V}$ .

(a)

Let $\mathcal{U}$ represent an $r$ -dimensional subspace of $\mathcal{V}$ , and suppose that $\mathcal{U}$ is invariant relative to $T$ . Show that there exists a basis $B$ for $\mathcal{V}$ such that the matrix representation of $T$ with respect to $B$ and $B$ is of the (upper block-triangular) form $\begin{pmatrix} \mathbf{E} & \mathbf{F} \\ \mathbf{0} & \mathbf{H} \end{pmatrix}$ (where $\mathbf{E}$ is of dimensions $r \times r$ ).

(b)

Let $\mathcal{U}$ and $\mathcal{W}$ represent subspaces of $\mathcal{V}$ such that $\mathcal{U} \oplus \mathcal{W} = \mathcal{V}$ (i.e., essentially disjoint subspaces of $\mathcal{V}$ whose sum is $\mathcal{V}$ ). Suppose that both $\mathcal{U}$ and $\mathcal{W}$ are invariant relative to $T$ . Show that there exists a basis $B$ for $\mathcal{V}$ such that the matrix representation of $T$ with respect to $B$ and $B$ is of the (block-diagonal) form $\diag (\mathbf{E}, \mathbf{H})$ [where the dimensions of $\mathbf{E}$ equal $\dim (\mathcal{W})$ ].

Problem 5C.6 from Axler S. "Linear Algebra Done Right" [Axl24]More problems in Section 5C "Upper-Triangular Matrices"

Suppose $\mathbf{F} = \mathbb {C}, V$ is finite-dimensional, and $T \in \mathcal{L}(V)$ . Prove that if $k \in \left\{ 1, \ldots , \operatorname {dim} V\right\}$ , then $V$ has a $k$ -dimensional subspace invariant under $T$ .

Problem 5C.10 from Axler S. "Linear Algebra Done Right" [Axl24]More problems in Section 5C "Upper-Triangular Matrices"

(a)(b)(c)

Suppose $T \in \mathcal{L}(V)$ and $v_{1}, \ldots , v_{n}$ is a basis of $V$ . Show that the following are equivalent.

(a)

The matrix of $T$ with respect to $v_{1}, \ldots , v_{n}$ is lower triangular.

(b)

$\operatorname {span}\left(v_{k}, \ldots , v_{n}\right)$ is invariant under $T$ for each $k = 1, \ldots , n$ .

(c)

$T v_{k} \in \operatorname {span}\left(v_{k}, \ldots , v_{n}\right)$ for each $k = 1, \ldots , n$ .

Remark.

A square matrix is called lower triangular if all entries above the diagonal are $0$ .

Problem 24.4 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 24.1 "Eigenvectors and eigenvalues (24.1-24.65). Solve problems that do not explicitly mention eigenvectors and eigenvalues as if they were problems on eigenvectors and eigenvalues."

What form does the matrix of a linear transformation have, if the first $k$ basis vectors are its eigenvectors?

Problem 24.72 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 24.2 "Invariant subspaces. Permutable transformations (24.66-24.114)"

(1)(2)

What form does the matrix of a linear transformation of an $n$ -dimensional vector space have, if the basis of an invariant subspace is formed by:

(1)

the first $k$ basis vectors

(2)

the last $n - k$ basis vectors?

Problem 24.100 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 24.2 "Invariant subspaces. Permutable transformations (24.66-24.114)"

(1)(2)

(1)

Let a linear transformation $\varphi$ of an $n$ -dimensional vector space $\mathcal{L}$ possess a chain of pairwise distinct invariant subspaces nested in one another: $\mathcal{L}_{1} \subset \mathcal{L}_{2} \subset \ldots \subset \mathcal{L}_{n} = \mathcal{L}$ . Prove that there exists in $\mathcal{L}$ a basis in which the matrix of the transformation $\varphi$ is upper triangular.

(2)

Let, in the basis $e_{1}, \ldots , e_{n}$ , the matrix of a linear transformation $\varphi$ of the space $\mathcal{L}$ be upper triangular. Prove that the subspaces $\mathcal{L}_{k} = \mathcal{L}\left\{ e_{1}, \ldots , e_{k}\right\} \quad (k = 1, \ldots , n)$ are invariant with respect to $\varphi$ and $\mathcal{L}_{k} \subset \mathcal{L}_{k + 1} \quad (k = 1, \ldots , n - 1)$ .

Problem 24.101 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 24.2 "Invariant subspaces. Permutable transformations (24.66-24.114)"

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

A linear transformation of the space $\mathcal{R}_{3}$ is given by the matrix $A$ in the standard basis. Reduce the matrix of the transformation to triangular form, if:

(1)

$A = \mathbf{A}_{241}$ . $\mathbf{A}_{241} = \left[\begin{smallmatrix} 6 & -1 & 1 \\ 5 & -5 & 5 \\ 4 & -9 & 9 \end{smallmatrix}\right]$ .

(2)

$A = \mathbf{A}_{222}$ . $\mathbf{A}_{222} = \left[\begin{smallmatrix} 1 & 1 & 1 \\ 2 & 2 & 2 \\ -3 & -3 & -3 \end{smallmatrix}\right]$ .

(3)

$A = \mathbf{A}_{283}$ . $\mathbf{A}_{283} = \left[\begin{smallmatrix} 2 & 5 & 1 \\ -1 & -3 & 0 \\ -2 & -3 & -2 \end{smallmatrix}\right]$ .

(4)

$A = \mathbf{A}_{263}$ . $\mathbf{A}_{263} = \left[\begin{smallmatrix} 3 & -3 & 1 \\ 3 & -2 & 2 \\ -1 & 2 & 0 \end{smallmatrix}\right]$ .

Problem 24.102 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 24.2 "Invariant subspaces. Permutable transformations (24.66-24.114)"

(1)(2)

(1)

Let $\mathcal{L}_{1} \subset \mathcal{L}_{2} \subset \ldots \subset \mathcal{L}_{r} = \mathcal{L}$ be a chain of subspaces of a vector space $\mathcal{L}$ , invariant with respect to a linear transformation $\varphi , \operatorname {dim} \mathcal{L}_{i} = n_{i}\left(n_{1} < n_{2} < \ldots \ldots < n_{r} = n\right)$ . Suppose the basis $e_{1}, \ldots , e_{n}$ is chosen so that the vectors $e_{1}, \ldots , e_{n_{i}}$ belong to $\mathcal{L}_{i}(i = 1, \ldots , r)$ . Show that the matrix $A_{\varphi }$ is upper block-triangular with diagonal blocks of size $k_{i} \times k_{i}$ , where $k_{i} = n_{i} - n_{i - 1}(i = 2, \ldots , r)$ , $k_{1} = n_{1}$ .

(2)

Let, in some basis, the matrix of a linear transformation be upper block-triangular. Prove that the transformation possesses a chain of invariant subspaces. Express their dimensions in terms of the orders of the diagonal blocks.

Problem 40.25 from Кострикин А.И. "Сборник задач по алгебре" [Кос15]More problems in Section 40 "Eigenvectors, invariant subspaces, root subspaces"

Prove that in an $n$ -dimensional complex space every linear operator has an invariant subspace of dimension $n-1$ .

Problem 40.43 from Кострикин А.И. "Сборник задач по алгебре" [Кос15]More problems in Section 40 "Eigenvectors, invariant subspaces, root subspaces"

Let $\mathcal{A}$ be a linear operator on a finite-dimensional real space $V$ . Prove that $V$ has a basis in which the matrix of the operator $\mathcal{A}$ has the block upper-triangular form

\left[\begin{smallmatrix} \boxed {A_{1}} & & & * \\ & \boxed {A_{2}} & & \\ & & \ddots & \\ 0 & & & \boxed {A_{n}} \end{smallmatrix}\right],

where the square blocks $A_{1}, \ldots , A_{n}$ have size at most 2.

Problem 73.12 from Кострикин А.И. "Сборник задач по алгебре" [Кос15]More problems in Section 73 "Initial information on representations of continuous groups"

(a)(b)

Let $G$ be a connected solvable topological group and $\rho$ a continuous homomorphism of $G$ into the group of nonsingular linear operators on a finite-dimensional complex space $V$ .

Prove that:

(a)

$V$ has a nonzero vector that is an eigenvector for all the operators $\rho (g), g \in G$ ;

(b)

$V$ has a basis $e_{1}, \ldots , e_{n}$ such that all the matrices $\rho (g)$ , $g \in G$ , are upper triangular in this basis.

Problem 73.13 from Кострикин А.И. "Сборник задач по алгебре" [Кос15]More problems in Section 73 "Initial information on representations of continuous groups"

Let $F$ be an algebraically closed field and $G$ a solvable group of nonsingular linear operators on a finite-dimensional vector space $V$ over $F$ . Prove that there exist a basis $e_{1}, \ldots , e_{n}$ of $V$ and a normal subgroup $N$ of $G$ of finite index (depending only on $n$ ) such that $N$ consists of upper triangular matrices.

Problem 4.9.4 from Meyer C.D. "Matrix Analysis and Applied Linear Algebra" [Mey00]More problems in Section 4.9 "Invariant Subspaces"

(a)(b)(c)(d)

Let $\mathbf{T}$ and $\mathbf{Q}$ be the matrices

\mathbf{T} = \begin{pmatrix} -2 & -1 & -5 & -2 \\ -9 & 0 & -8 & -2 \\ 2 & 3 & 11 & 5 \\ 3 & -5 & -13 & -7 \end{pmatrix} \quad \text{and} \quad \mathbf{Q} = \begin{pmatrix} 1 & 0 & 0 & -1 \\ 1 & 1 & 3 & -4 \\ -2 & 0 & 1 & 0 \\ 3 & -1 & -4 & 3 \end{pmatrix}.

(a)

Explain why the columns of $\mathbf{Q}$ are a basis for $\mathbb {R}^{4}$ .

(b)

Verify that $\mathcal{X} = \operatorname {span}\left\{ \mathbf{Q}_{*1}, \mathbf{Q}_{*2}\right\}$ and $\mathcal{Y} = \operatorname {span}\left\{ \mathbf{Q}_{*3}, \mathbf{Q}_{*4}\right\}$ are each invariant subspaces under $\mathbf{T}$ .

(c)

Describe the structure of $\mathbf{Q}^{-1}\mathbf{T}\mathbf{Q}$ without doing any computation.

(d)

Now compute the product $\mathbf{Q}^{-1}\mathbf{T}\mathbf{Q}$ to determine $[\mathbf{T}_{/\mathcal{X}}]_{\left\{ \mathbf{Q}_{*1}, \mathbf{Q}_{*2}\right\} }$ and $[\mathbf{T}_{/\mathcal{Y}}]_{\left\{ \mathbf{Q}_{*3}, \mathbf{Q}_{*4}\right\} }$ .

Problem 4.9.5 from Meyer C.D. "Matrix Analysis and Applied Linear Algebra" [Mey00]More problems in Section 4.9 "Invariant Subspaces"

Let $\mathbf{T}$ be a linear operator on a space $\mathcal{V}$ , and suppose that

\mathcal{B} = \left\{ \overrightarrow {u}_1, \ldots , \overrightarrow {u}_r, \; \overrightarrow {w}_1, \ldots , \overrightarrow {w}_q\right\}

is a basis for $\mathcal{V}$ such that $[\mathbf{T}]_{\mathcal{B}}$ has the block-diagonal form

[\mathbf{T}]_{\mathcal{B}} = \begin{pmatrix} \mathbf{A}_{r \times r} & \mathbf{0} \\ \mathbf{0} & \mathbf{C}_{q \times q} \end{pmatrix}.

Explain why $\mathcal{U} = \operatorname {span}\left\{ \overrightarrow {u}_1, \ldots , \overrightarrow {u}_r\right\}$ and $\mathcal{W} = \operatorname {span}\left\{ \overrightarrow {w}_1, \ldots , \overrightarrow {w}_q\right\}$ must each be invariant subspaces under $\mathbf{T}$ .

Problem 4.9.6 from Meyer C.D. "Matrix Analysis and Applied Linear Algebra" [Mey00]More problems in Section 4.9 "Invariant Subspaces"

If $\mathbf{T}_{n \times n}$ and $\mathbf{P}_{n \times n}$ are matrices such that

\mathbf{P}^{-1}\mathbf{T}\mathbf{P} = \begin{pmatrix} \mathbf{A}_{r \times r} & \mathbf{0} \\ \mathbf{0} & \mathbf{C}_{q \times q} \end{pmatrix},

explain why $\mathcal{U} = \operatorname {span}\left\{ \mathbf{P}_{*1}, \ldots , \mathbf{P}_{*r}\right\}$ and $\mathcal{W} = \operatorname {span}\left\{ \mathbf{P}_{*r+1}, \ldots , \mathbf{P}_{*n}\right\}$ are each invariant subspaces under $\mathbf{T}$ .

Recall (Example 4.8.3): if $\mathbf{T}$ is regarded as the linear operator on $\mathbb {R}^{n \times 1}$ defined by matrix multiplication $\mathbf{T}(\overrightarrow {x}) = \mathbf{T}\overrightarrow {x}$ , then with respect to the standard basis $\mathcal{S}$ , $[\mathbf{T}]_{\mathcal{S}} = \mathbf{T}$ .

Problem 5.9.15 from Meyer C.D. "Matrix Analysis and Applied Linear Algebra" [Mey00]More problems in Section 5.9 "Complementary Subspaces"

For complementary subspaces $X$ and $Y$ of $\mathbb {R}^{n}$ , let $P$ be the projector onto $X$ along $Y$ , and let $Q = \left[X\mid Y\right]$ , where the columns of $X$ and $Y$ constitute bases for $X$ and $Y$ , respectively. Prove that if $Q^{-1}A_{n\times n}Q$ is partitioned as $Q^{-1}AQ = \begin{pmatrix} A_{11}& A_{12} \\ A_{21}& A_{22} \end{pmatrix}$ , then

Q\begin{pmatrix} A_{11}& 0 \\ 0& 0 \end{pmatrix}Q^{-1} = PAP, \quad Q\begin{pmatrix} 0& A_{12} \\ 0& 0 \end{pmatrix}Q^{-1} = PA(I-P), \quad Q\begin{pmatrix} 0& 0 \\ A_{21}& 0 \end{pmatrix}Q^{-1} = (I-P)AP, \quad Q\begin{pmatrix} 0& 0 \\ 0& A_{22} \end{pmatrix}Q^{-1} = (I-P)A(I-P).

Remark.

This means that if $A$ is a linear operator on $\mathbb {R}^{n}$ , and $B = B_{X}\cup B_{Y}$ (the bases making up the columns of $X,Y$ ), then the matrix representation of $A$ with respect to $B$ is $[A]_{B} = \begin{pmatrix} A_{11}& A_{12}\\ A_{21}& A_{22} \end{pmatrix}$ , where $A_{11} = \left[PA P/X\right]_{B_{X}}$ , $A_{12} = \left[PA(I-P)/Y\right]_{B_{Y}B_{X}}$ , $A_{21} = \left[(I-P)AP/X\right]_{B_{X}B_{Y}}$ , $A_{22} = \left[(I-P)A(I-P)/Y\right]_{B_{Y}}$ (each a matrix representation of a restricted operator).

Problem 1127 from Смирнов Ю.М. "Сборник задач по аналитической геометрии и линейной алгебре" [Сми16]More problems in Section 11.2 "Kernel and Image of a Linear Operator. Invariant Subspaces. Projectors. Complexification and Realification"

(1)(2)

Characterize the matrices of operators А in the basis $e_{1}, \ldots , e_{n}$ such that

(1)

the subspace $\left\langle e_{1}, \ldots , e_{k}\right\rangle$ is invariant under $\mathbf{A}$ ;

(2)

the subspace $\left\langle e_{k+1}, \ldots , e_{n}\right\rangle$ is invariant under $\mathbf{A}$ .

Problem 1182 from Смирнов Ю.М. "Сборник задач по аналитической геометрии и линейной алгебре" [Сми16]More problems in Section 11.4 "Eigenvalues, Eigenvectors"

Prove that a linear operator on a space $V$ of dimension $n$ over an algebraically closed field always has an invariant subspace of dimension $n-1$ .

Problem 1185 from Смирнов Ю.М. "Сборник задач по аналитической геометрии и линейной алгебре" [Сми16]More problems in Section 11.4 "Eigenvalues, Eigenvectors"

Show that in a space $V$ of dimension $n$ over an algebraically closed field every operator А has invariant subspaces of all dimensions from 0 to $n$ .

References

[Har01] Harville D.A. Matrix Algebra: Exercises and Solutions. Springer, 2001.

[Axl24] Axler S. Linear Algebra Done Right. 4th ed., Springer, 2024.

[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.

[Mey00] Meyer C.D. Matrix Analysis and Applied Linear Algebra. SIAM, 2000.

[Сми16] Смирнов Ю.М. Сборник задач по аналитической геометрии и линейной алгебре. МЦНМО, 2016.