Linear algebra prob

Question

let A = Find an orthogonal matrix P with rational entries and a diagonal matrix D such that D=P^TAP.

Answer 1

There are a couple of things we should start to note here: The matrix A is symmetric. A rational number is any number x such that x = a/b for some integers a and b. Let's get started with our solution. The first step is to find the eigenvalues and eigenvectors of A. For any eigenvector v ≠ 0 and eigenvalue λ, we have that A * v = λ * v. This is equivalent to saying that (A - λ * I) * v = 0, which means λ is a root of the characteristic polynomial of A.

To find the eigenvalues, we compute the determinant of (A - λ * I). That is, we compute:

| 25 - λ -4 -4 |

| -4 19 - λ -8 |

| -4 -8 19 - λ |

We expand this determinant along the first row:

= (25 - λ) * ((19 - λ)^2 - 64)

1. 4 * (-4 * (19 - λ) - 32)
2. 4 * (32 + 4 * (19 - λ))

From here, we simplify the expression to find a cubic polynomial in λ, factor it, and solve for the eigenvalues. Once we know the eigenvalues, we can find corresponding eigenvectors and construct an orthogonal matrix P with rational entries such that D = P^T A P, where D is a diagonal matrix of eigenvalues.