Question

If $\lambda_1$ and $\lambda_2$ are two distinct eigen values of a symmetric matrix, $X_1$ and $X_2$ are the eigen vectors corresponding to $\lambda_1, \lambda_2$ respectively, then

• A. $X_1 + X_2 = 0$
• B. $X_1^T X_2 = 1$
• C. $X_1 - X_2 = 0$
• D. $X_1^T X_2 X_1 = X_2^T X_1 X_2$

Hint:

• For a symmetric matrix, eigenvectors corresponding to distinct eigenvalues are orthogonal.
• Orthogonality of vectors $X_1$ and $X_2$ means their dot product (or inner product) is zero: $X_1^T X_2 = 0$.
• Evaluate each option based on this fundamental property.
• If $X_1^T X_2 = 0$, then any scalar multiple of $X_1$ or $X_2$ by this zero scalar will result in a zero vector or zero scalar, making statements like option (d) true if interpreted as vector equality $\mathbf{0}=\mathbf{0}$.

Answer 1

The Correct Option is D

Step 1: Recall the properties of symmetric matrices

• A symmetric matrix A satisfies A = A^T.
• Eigenvalues of a symmetric matrix are real numbers.
• Eigenvectors corresponding to distinct eigenvalues of a symmetric matrix are orthogonal to each other.

Step 2: Understand what orthogonality means here

If two vectors X₁ and X₂ are orthogonal, then their dot product is zero:

$$ X_1^{\mathrm{T}} X_2 = 0 $$

Since λ₁ and λ₂ are distinct, the corresponding eigenvectors X₁ and X₂ are orthogonal:

$$ X_1^{\mathrm{T}} X_2 = 0 $$

Step 3: Analyze each option

1. Option 1: $$ X_1 + X_2 = 0 $$

This means the sum of the two eigenvectors is zero, which implies X₁ = -X₂. There is no general reason for this to be true for eigenvectors of distinct eigenvalues, so this option is incorrect.

1. Option 2: $$ X_1^{\mathrm{T}} X_2 = 1 $$

This states that the dot product of the two eigenvectors is 1. Since the eigenvectors are orthogonal, their dot product must be zero, so this option is incorrect.

1. Option 3: $$ X_1 - X_2 = 0 $$

This implies that the two eigenvectors are equal, which contradicts the property that eigenvectors of distinct eigenvalues are linearly independent and distinct. So, this is incorrect.

1. Option 4: $$ X_1^{\mathrm{T}} X_2 X_1 = X_2^{\mathrm{T}} X_1 X_2 $$

Let's analyze this expression carefully:

• Note: Here, \(X_1^{\mathrm{T}} X_2 and X_2^{\mathrm{T}} X_1\) are scalars (single numbers) because these are dot products of vectors.
• Since dot products are commutative for real vectors, we have:
• Therefore:
• Since the scalars \(X_1^{\mathrm{T}} X_2 and X_2^{\mathrm{T}} X_1\) are equal, the difference depends on the vectors \(X_1\) and \(X_2\).
• Because \(X_1^{\mathrm{T}} X_2 = 0\) (from orthogonality), both sides are zero vectors:
• Hence, the equality holds true

For a symmetric matrix, eigenvectors corresponding to distinct eigenvalues are orthogonal. 
Therefore, the dot product is zero, which makes option 4 true while the others are false.