Let A be an n x n matrix over C. An n x n matrix B is called a square root of A if B² = A. Find the square roots of the 2 × 2 identity matrix applying the spectral theorem. The eigenvalues of I2 are A₁ = 1 and ₂ = 1. As normalized eigenvectors choose ei eidi cos(0) ei sin(0) (6)), (te: sin(0) -ei which form an orthonormal basis in C². Four cases (√√₁, √√₂) = (1,1), (√√√₁, √√√₂) = (1,−1), (√√₁,√√√₂) = (-1, 1), (√√₁, √√√₂)=(−1, −1) have to be studied. The first and last cases are trivial. So study the second case (√√₁,√√₂) = (1, -1). The second case and the third case are "equivalent".
Let A be an n x n matrix over C. An n x n matrix B is called a square root of A if B² = A. Find the square roots of the 2 × 2 identity matrix applying the spectral theorem. The eigenvalues of I2 are A₁ = 1 and ₂ = 1. As normalized eigenvectors choose ei eidi cos(0) ei sin(0) (6)), (te: sin(0) -ei which form an orthonormal basis in C². Four cases (√√₁, √√₂) = (1,1), (√√√₁, √√√₂) = (1,−1), (√√₁,√√√₂) = (-1, 1), (√√₁, √√√₂)=(−1, −1) have to be studied. The first and last cases are trivial. So study the second case (√√₁,√√₂) = (1, -1). The second case and the third case are "equivalent".
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
Related questions
Question
![Let A be an n x n matrix over C. An n x n matrix B is called
a square root of A if B² = A. Find the square roots of the 2 × 2 identity matrix
applying the spectral theorem. The eigenvalues of I2 are A₁ = 1 and ₂ = 1. As
normalized eigenvectors choose
eidi cos(
ei sin(0)
ei sin(0)
-ei cos(0)
which form an orthonormal basis in C².
Four cases
(√₁,√√√₂) = (1, 1),
(√√√₁, √√√√₂) = (1, −1),
(√√₁, √√₂) = (-1, 1),
(√√√₁,√√√₂)=(−1, −1)
have to be studied. The first and last cases are trivial. So study the second case
(√√1, √√₂) = (1, -1). The second case and the third case are "equivalent".](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F0feac8dc-2b65-4c46-868a-03ceb7cf437c%2F2d6e1c7e-dc84-4659-b279-6bab99445591%2Ft2ulsj9_processed.jpeg&w=3840&q=75)
Transcribed Image Text:Let A be an n x n matrix over C. An n x n matrix B is called
a square root of A if B² = A. Find the square roots of the 2 × 2 identity matrix
applying the spectral theorem. The eigenvalues of I2 are A₁ = 1 and ₂ = 1. As
normalized eigenvectors choose
eidi cos(
ei sin(0)
ei sin(0)
-ei cos(0)
which form an orthonormal basis in C².
Four cases
(√₁,√√√₂) = (1, 1),
(√√√₁, √√√√₂) = (1, −1),
(√√₁, √√₂) = (-1, 1),
(√√√₁,√√√₂)=(−1, −1)
have to be studied. The first and last cases are trivial. So study the second case
(√√1, √√₂) = (1, -1). The second case and the third case are "equivalent".
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