Let α, β be two non-zero real numbers and v1, v2 be two non-zero real vectors of size 3 × 1. Suppose that v1 and v2 satisfy...

Question 37

Let α, β be two non-zero real numbers and v₁, v₂ be two non-zero real vectors of size 3 × 1. Suppose that v₁ and v₂ satisfy v₁^Tv₂ = 0, v₁^Tv₁ = 1, and v₂^Tv₂ = 1. Let A be the 3 × 3 matrix given by:

A = αv₁v₁^T + βv₂v₂^T

The eigenvalues of A are ________.

(A) 0, α, β

(B) 0, α + β, α - β

(C) 0, (α + β)/2, √(αβ)

(D) 0, 0, √(α² + β²)

Solution

To solve for the eigenvalues of matrix A, we test the given vectors as potential eigenvectors:

1. Testing v₁: Multiplying A by v₁:
   Av1 = (αv1v1T + βv2v2T)v1 = αv1(v1Tv1) + βv2(v2Tv1)
   Using the given conditions (v₁^Tv₁ = 1 and v₂^Tv₁ = 0), we get:
   Av1 = αv1(1) + βv2(0) = αv1
   Thus, α is an eigenvalue.


2. Testing v₂: Similarly, multiplying A by v₂:
   Av2 = αv1(v1Tv2) + βv2(v2Tv2) = αv1(0) + βv2(1) = βv2
   Thus, β is an eigenvalue.


3. Finding the Third Eigenvalue: Since v₁ and v₂ are in ℝ³, there exists a vector v₃ orthogonal to both.
   Av3 = αv1(0) + βv2(0) = 0
   Thus, 0 is the third eigenvalue.

Correct Answer: (A) 0, α, β

GATE EC Previous Year Papers with Solutions →

Electronics and Communication Study Material →