1+i i 2i1 -i 2 C: An example of diagonalisation : Let T be the operator of C³ with matrix M = ( 7 0 1) Find 3 vectors e₁ e2 e3 satisfying the equations : T(e₁) = e₁ T(e₂) = ie2 T(e3-ie3
1+i i 2i1 -i 2 C: An example of diagonalisation : Let T be the operator of C³ with matrix M = ( 7 0 1) Find 3 vectors e₁ e2 e3 satisfying the equations : T(e₁) = e₁ T(e₂) = ie2 T(e3-ie3
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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![**Example of Diagonalization:**
Let \( T \) be the operator of \( \mathbb{C}^3 \) with matrix:
\[ M = \begin{pmatrix}
1 & 2 & 1 + i \\
0 & i & 2i - 1 \\
0 & 0 & -i
\end{pmatrix} \]
**Task:**
1. Find 3 vectors \( e_1 \), \( e_2 \), and \( e_3 \) satisfying the equations:
\[ \begin{cases}
T(e_1) = e_1 \\
T(e_2) = ie_2 \\
T(e_3) = -ie_3
\end{cases} \]
This example illustrates the process of diagonalization. The given matrix \( M \) is a complex matrix. Your task is to find three vectors \( e_1 \), \( e_2 \), and \( e_3 \) that satisfy the transformation properties shown above.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F46a741e2-1f18-40f1-9eda-db0ce81998cf%2F018a8dab-6449-4cde-b1cd-335edc79e08d%2F3wub8vg_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Example of Diagonalization:**
Let \( T \) be the operator of \( \mathbb{C}^3 \) with matrix:
\[ M = \begin{pmatrix}
1 & 2 & 1 + i \\
0 & i & 2i - 1 \\
0 & 0 & -i
\end{pmatrix} \]
**Task:**
1. Find 3 vectors \( e_1 \), \( e_2 \), and \( e_3 \) satisfying the equations:
\[ \begin{cases}
T(e_1) = e_1 \\
T(e_2) = ie_2 \\
T(e_3) = -ie_3
\end{cases} \]
This example illustrates the process of diagonalization. The given matrix \( M \) is a complex matrix. Your task is to find three vectors \( e_1 \), \( e_2 \), and \( e_3 \) that satisfy the transformation properties shown above.
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