Need Help Understanding and Solving Matrix Factorization Problems

Question

I hope you're doing well. I'm currently working on a set of matrix factorization problems and could really use some assistance in understanding and solving them. I've been trying to make sense of the concepts involved, but I've hit a bit of a roadblock. I'm reaching out to this knowledgeable community in the hopes that someone could provide some guidance and insights.

The specific problems I'm tackling involve diagonalizing matrices and finding eigenvalues and eigenvectors. While I have a basic understanding of these concepts, I'm struggling to grasp the finer details and how to apply them correctly.

Here are the two problems I'm trying to work through:

Problem 1: Given a matrix (A) that can be factored as (A = PDP^{-1}), where: [ A = \begin{bmatrix} 1 & 0 & -2 \\ -1 & 2 & -2 \\ 1 & -3 & 1 \end{bmatrix} ] and [ D = \begin{bmatrix} 2 & 0 & 0 \\ 0 & -1 & 0 \\ 0 & 0 & 3 \end{bmatrix} ]

(a) I need to show that the characteristic polynomial for (A) can be written as ((\lambda - 2)(\lambda + 1)(\lambda - 3)).

In my attempt to solve this part, I noticed that (D) is a diagonal matrix with its diagonal elements being the eigenvalues of matrix (A). Given that (D) consists of eigenvalues 2, -1, and 3, I'm trying to understand how these eigenvalues are related to the factorization (A = PDP^{-1}) and how they lead to the characteristic polynomial ((\lambda - 2)(\lambda + 1)(\lambda - 3)). However, I'm not entirely clear on the steps to bridge this connection. Could someone provide some guidance on how to establish this link and derive the characteristic polynomial based on the given information?

(b) I also need to calculate the matrix (P).

In my approach to solving this part, I'm aware that the columns of matrix (P) correspond to the eigenvectors of matrix (A). For the eigenvalues 2, -1, and 3, I've already found the corresponding eigenvectors: [0, 1, 1], [1, 1, 0], and [1, 2, 1] respectively. However, I'm unsure how to proceed to assemble these eigenvectors into matrix (P). Could someone please provide a step-by-step explanation of how to construct matrix (P) using the given eigenvectors?

Answer 1

I'll provide a sketch.

For a), you'll need to factor cleverly. As $A$ and $D$ are similar matrices, we have $A = PDP^{-1}.$ Now, we need to factor $PDP^{-1} - \lambda I$ in the following way:

\begin{align*} PDP^{-1} - \lambda I &= P(DP^{-1} - \lambda P^{-1} ) \\ & = P (D - \lambda I) P^{-1} \\ \end{align*}

and now use the fact that the determinant is multiplicative, i.e., $\det AB = (\det A)(\det B).$

For b), you already have it - the matrix $P$ is the matrix whose columns correspond to the eigenvectors of $A.$ Note that if $(\lambda_i, v_i)$ is an eigenpair of $A$, then column $i$ of $P$ corresponds to the eigenvector linked with $\lambda_i.$

Answer 2

It is well-known that if $A = PDP^{-1}$ for diagonal matrix $D$, then $D$ contains eigenvalues of $A$ on its diagonal, columns of $P$ are corresponding eigenvectors.

To prove this we rewrite $A = PDP^{-1}$ as $AP = PD$. Let $P = [\vec v_1 \ldots \vec v_n]$, $D = diag\{\alpha_1, \ldots, \alpha_n\}$. So, $AP=PD$ leads to $A\vec v_i = \alpha_i \vec v_i$, so $\vec v_i$ are eigenvectors and $\alpha_i$ are eigenvalues.