Problem of the week - A simple question about eigenvalues

There exist multiple ways to decompose or factorize a matrix in Linear Algebra, some of them have been already dealt in previous problems like the LU decomposition or the rank factorization, which is done implicitly in the process of transforming a matrix to row echelon form. One of the most used matrix decompositions is the eigendecomposition, which is related to the concept of diagonalization, and decomposes a matrix into eigenvectors and eigenvalues.

Eigendecomposition plays a key role in computer vision and machine learning in general. Well known examples are PCA (Principal Component Analysis) for dimensionality reduction or EigenFaces for face recognition. As another important example of the use of this decomposition, Google, relies upon eigenvalues and eigenvectors to rank pages with respect to relevance.

From the perspective of Linear transformations, this decomposition just finds the vectors which images keep the same direction, the eigenvectors, and their scaling factors, the eigenvalues.

Although this problem of the week doesn't request to find the eigendecomposition of a matrix, it is related to the concept of eigenvalues, and to solve it, you will need to know the foundation of the procedures used to compute them.

Give all values of x for which A has an eigenvalue equal to -3.

    ┌            ┐    
    │  1  -6   x │
A = │ 2x  -9   2 │
    │ -3   3  -4 │
    └            ┘

The matrix A will have -3 as an eigenvalue if and only if det(A + 3I) = 0.

Therefore, to find the values of x, we need to solve that equation.

│ 1+3    -6     x │
│  2x  -9+3     2 │ = 0
│  -3     3  -4+3 │


│  4  -6   x │
│ 2x  -6   2 │ = 24 + 36 + 6x2 - 18x - 24 - 12x = 0
│ -3   3  -1 │

6x2 - 30x + 36 = 0

x2 - 5x + 6 = 0

(x - 2)(x - 3) = 0

x = 2 or x = 3

Hence, A has -3 as an eigenvalue if and only if x equals 2 or 3.