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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
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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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".
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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