1. What is an Eigenvector?

An eigenvector of a square matrix is a non-zero vector that only changes by a scalar factor (the eigenvalue) when that matrix is applied to it. For matrix A and eigenvector x with eigenvalue λ, the relationship is:

2. How to Find Eigenvectors

To find eigenvectors for a given eigenvalue λ:

Where:

• \( A \) — Original square matrix
• \( \lambda \) — Eigenvalue
• \( I \) — Identity matrix of same size as A
• \( \mathbf{x} \) — Eigenvector to be found

Steps:

1. Subtract λ from each diagonal element of A (creating A - λI)
2. Solve the resulting homogeneous system of linear equations
3. The non-trivial solutions are the eigenvectors

3. Importance of Eigenvectors

Applications: Eigenvectors are fundamental in many areas including stability analysis, quantum mechanics, facial recognition (PCA), vibration analysis, and more.

4. Using the Calculator

Instructions: Enter all 9 elements of your 3×3 matrix and the known eigenvalue. The calculator will solve (A - λI)x = 0 to find the corresponding eigenvector.

5. Frequently Asked Questions (FAQ)

Q1: Can a matrix have multiple eigenvectors for one eigenvalue?
A: Yes, this is called the geometric multiplicity of the eigenvalue.

Q2: What if my matrix has complex eigenvalues?
A: The calculator works with real numbers only. Complex eigenvalues would produce complex eigenvectors.

Q3: How do I know if my eigenvector is correct?
A: Multiply your matrix by the eigenvector - the result should be the eigenvector scaled by the eigenvalue.

Q4: What if I get the zero vector as a result?
A: Eigenvectors must be non-zero. This may indicate an error in your eigenvalue or that the calculator couldn't find a solution.

Q5: Are eigenvectors unique?
A: No, any scalar multiple of an eigenvector is also an eigenvector for the same eigenvalue.