MAT223 Lecture 10

Completed Notes Status

Central objective: Understand the process of finding eigenvalues and eigenvectors, and analyze the relationship between matrix invertibility and diagonalizability.
Key concepts:
- Eigenvalue and Eigenvector computation: Finding roots of the Characteristic Polynomial $c_{A} (x) = det (x I_{n} - A)$ and solving the Homogeneous System $(A - λ I_{n}) \vec{x} = \vec{0}$ .
- Matrix Diagonalization ( $A = P D P^{- 1}$ ): The order of eigenvalues in the diagonal matrix $D$ must match the order of corresponding basic eigenvectors in matrix $P$ .
- Matrix Invertibility: A matrix is invertible if and only if $rank (A) = n$ , $det A \neq 0$ , and the homogeneous system $A \vec{x} = \vec{0}$ has only the trivial solution.
Connections:
- There is no direct connection between a matrix being a Diagonalizable Matrix and being invertible; a matrix can be one, both, or neither.
- A square matrix is non-invertible if and only if $0$ is an Eigenvalue of the matrix.

#tk: idk the above, doesn't make sense. (Regarding why a matrix having 0 as an eigenvalue means it is non-invertible, and vice-versa).
- Answer: A matrix $C$ is non-invertible if and only if its determinant $det (C) = 0$ . The characteristic equation is $det (C - λ I_{n}) = 0$ . If we plug in $λ = 0$ , we get $det (C - 0 I_{n}) = det (C)$ . Therefore, if $det (C) = 0$ , then $λ = 0$ is a valid root (eigenvalue). Conversely, if $λ = 0$ is an eigenvalue, then $det (C - 0 I_{n}) = det (C) = 0$ , which proves $C$ is non-invertible.

Remember/Understand:
- What is the required relationship between the matrices $D$ and $P$ when diagonalizing a matrix $A$ as $A = P D P^{- 1}$ ?
- What must be true about the eigenvalues of a matrix if the matrix is known to be non-invertible?
Apply/Analyze:
- Given an upper-triangular $3 \times 3$ matrix, how can you immediately determine its characteristic polynomial and eigenvalues?
- If a $3 \times 3$ matrix has distinct eigenvalues $λ = 1, 3, - 4$ , prove whether it is necessarily diagonalizable.
Evaluate/Create:
- Construct an example of a $2 \times 2$ matrix that is invertible but not diagonalizable, and prove why it fails to be diagonalizable.

Matrix Invertibility vs. Diagonalizability:
- Why it's challenging: It is easy to assume that a matrix failing to have full rank (non-invertible) also fails to be diagonalizable, but the two properties are entirely independent.
- Study strategy: Memorize counter-examples for each case (e.g., the identity matrix is both; a shear matrix $[\begin{matrix} 1 & 1 \\ 0 & 1 \end{matrix}]$ is invertible but not diagonalizable; a zero matrix is diagonalizable but not invertible).
Characteristic Polynomial of singular matrices:
- Why it's challenging: Understanding why $λ = 0$ forces the constant term of the characteristic polynomial to be zero, tying back to the determinant.
- Study strategy: Write out the characteristic polynomial $c_{C} (x) = det (x I_{n} - C)$ and evaluate it explicitly at $x = 0$ to see that it equals $(- 1)^{n} det (C)$ .

Immediate review actions:
- Review all [COMPLETED] insertions and verify with course materials
- Address all #tk items (if any)
- Re-read Lecture Summary and confirm key links
Practice and application:
- Answer Remember/Understand questions
- Answer Apply/Analyze questions
- Attempt Evaluate/Create questions
Deep dive study:
- Focus on Challenging Concepts (above)
- Extend atomic notes only where the new lecture adds content
Verification and integration:
- Cross-reference notes with textbook/slides
- Identify remaining gaps
- Link this lecture to prior/future lecture notes (lecture notes only; no MOCs)