Prove Triangularization of T and Nilpotency of Given Matrix

Math Problem Statement

Let T ∈ A_F(V). If T has all its characteristic roots in F, then show that there is a basis of V over F in which the matrix of T is triangular. Additionally, do any two of the following: (1) Prove that the given matrix is nilpotent and find its triangular form. (2) Prove that M_λ is S-invariant if S commutes with T.

Solution

Let's analyze the problem step by step.

Part A:

Statement:

Let $T \in A_F(V)$ (set of endomorphisms over vector space $V$ over $F$ ). If $T$ has all its characteristic roots in $F$ , then show that there exists a basis of $V$ over $F$ such that the matrix representation of $T$ in this basis is triangular.

Solution Outline:

Characteristic Roots in $F$ : If $T$ has all its characteristic roots in $F$ , its characteristic polynomial splits into linear factors in $F[x]$ . $p_T(x) = \prod_{i=1}^n (x - \lambda_i),$ where $\lambda_i$ are eigenvalues in $F$ .
Cyclic Subspaces: For each eigenvalue $\lambda$ , the generalized eigenspaces $V_{\lambda}$ corresponding to $\lambda$ are $T$ -invariant.
Jordan Normal Form or Schur Triangularization: Since $p_T(x)$ splits, $T$ is similar to a matrix in upper triangular form. By the Schur decomposition theorem, there exists a basis of $V$ such that $T$ has a triangular representation.

Conclusion: Triangularization follows from the properties of splitting fields and existence of eigenvectors.

Part B:

1. Prove that the given matrix is nilpotent and find its triangular form.

Matrix:

1 & 1 & 1 \\ -1 & -1 & -1 \\ 1 & 1 & 0 \end{bmatrix}$$ 1. **Check Nilpotency:** - Compute $$A^2$$: $$A^2 = A \cdot A = \begin{bmatrix} 1 & 1 & 1 \\ -1 & -1 & -1 \\ 1 & 1 & 0 \end{bmatrix} \begin{bmatrix} 1 & 1 & 1 \\ -1 & -1 & -1 \\ 1 & 1 & 0 \end{bmatrix}.$$ The result is: $$A^2 = \begin{bmatrix} 1 & 1 & 0 \\ -1 & -1 & 0 \\ 0 & 0 & 0 \end{bmatrix}.$$ - Compute $$A^3$$: $$A^3 = A \cdot A^2 = \begin{bmatrix} 1 & 1 & 1 \\ -1 & -1 & -1 \\ 1 & 1 & 0 \end{bmatrix} \begin{bmatrix} 1 & 1 & 0 \\ -1 & -1 & 0 \\ 0 & 0 & 0 \end{bmatrix} = \begin{bmatrix} 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{bmatrix}.$$ Since $$A^3 = 0$$, the matrix is nilpotent. 2. **Triangular Form**: - Perform Gaussian elimination or basis change to upper triangularize $$A$$. A possible triangular form is: $$T = \begin{bmatrix} 1 & 1 & 0 \\ 0 & -1 & 0 \\ 0 & 0 & 0 \end{bmatrix}.$$ --- #### **2. Prove the properties of $$M_\lambda$$:** Let $$T \in A_F(V)$$, $$\lambda \in F$$ (characteristic of $$T$$), and $$M_\lambda = \{ v \in V : Tv = \lambda v \}$$. If $$S \in A_F(V)$$ commutes with $$T$$, prove $$M_\lambda$$ is $$S$$-invariant. #### **Solution Outline**: 1. **Define $$v \in M_\lambda$$:** $$Tv = \lambda v.$$ 2. **Action of $$S$$:** Apply $$T$$ to $$Sv$$ and use commutativity: $$T(Sv) = S(Tv) = S(\lambda v) = \lambda(Sv).$$ Thus, $$Sv \in M_\lambda$$. **Conclusion**: $$M_\lambda$$ is $$S$$-invariant. --- ### Questions for Further Exploration: 1. How can triangularization simplify solving eigenvalue problems for $$T$$? 2. What properties of nilpotent matrices generalize to larger dimensions? 3. What is the significance of the Jordan form in analyzing nilpotent matrices? 4. Why does $$S$$-invariance simplify computations in the context of commuting operators? 5. How can Schur's triangularization theorem be extended to complex vector spaces? **Tip**: Always verify eigenvalues and eigenvectors explicitly when working with triangular matrices!

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Math Problem Analysis

Mathematical Concepts

Linear Algebra
Matrix Triangularization
Nilpotent Matrices
Eigenvalues and Eigenvectors

Formulas

Characteristic Polynomial: p_T(x) = det(xI - T)
Matrix Multiplication for Powers of Matrices
Nilpotent Matrix Condition: A^k = 0 for some positive integer k

Theorems

Schur's Triangularization Theorem
Properties of Eigenvectors
Commutativity and Invariance of Subspaces

Suitable Grade Level

Undergraduate - Advanced Linear Algebra

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