University Linear Algebra Quiz: Constructing P and D

📚 Quick Study Guide

• 🍎 Diagonalization: A square matrix $A$ is diagonalizable if there exists an invertible matrix $P$ and a diagonal matrix $D$ such that $A = PDP^{-1}$.
• 🔑 Finding P: The columns of $P$ are the eigenvectors of $A$. Make sure these eigenvectors are linearly independent.
• 💡Finding D: The diagonal entries of $D$ are the eigenvalues of $A$, corresponding to the eigenvectors in $P$. If the first column of $P$ is the eigenvector for eigenvalue $\lambda_1$, then the first entry of $D$ is $\lambda_1$.
• 📐Eigenvalues: To find eigenvalues, solve the characteristic equation $\text{det}(A - \lambda I) = 0$, where $I$ is the identity matrix.
• 🧮Eigenvectors: For each eigenvalue $\lambda$, find the eigenvectors by solving $(A - \lambda I)v = 0$, where $v$ is the eigenvector.
• 🧭 Order Matters: The order of eigenvalues and eigenvectors must be consistent. If you switch the order in $P$, you must switch the corresponding order in $D$.
• ⏱️ Invertibility of P: Matrix $P$ must be invertible; this happens when the eigenvectors are linearly independent and form a basis for the vector space.

🧪 Practice Quiz

1. Question 1: If $A = PDP^{-1}$, what are the columns of $P$?
   1. Eigenvalues of $A$
   2. Eigenvectors of $A$
   3. Rows of $A$
   4. Columns of $A$
2. Question 2: What type of matrix is $D$ in the equation $A = PDP^{-1}$?
   1. Identity Matrix
   2. Invertible Matrix
   3. Diagonal Matrix
   4. Zero Matrix
3. Question 3: How do you find the eigenvalues of matrix $A$?
   1. Solve $Av = 0$
   2. Solve $\text{det}(A - \lambda I) = 0$
   3. Find the trace of $A$
   4. Calculate $A^2$
4. Question 4: What condition must $P$ satisfy for $A = PDP^{-1}$ to hold true?
   1. $P$ must be orthogonal
   2. $P$ must be symmetric
   3. $P$ must be invertible
   4. $P$ must be diagonal
5. Question 5: If the first eigenvector in $P$ corresponds to eigenvalue $\lambda_1$, where should $\lambda_1$ appear in $D$?
   1. First row, second column
   2. Second row, first column
   3. First diagonal entry
   4. Last diagonal entry
6. Question 6: What happens if the eigenvectors used to form $P$ are linearly dependent?
   1. $A$ is still diagonalizable
   2. $P$ is not invertible
   3. $D$ becomes an identity matrix
   4. $A$ becomes a zero matrix
7. Question 7: If $A$ is a 3x3 matrix and has only two distinct eigenvalues, is $A$ diagonalizable?
   1. Yes, always
   2. No, never
   3. Only if the eigenspace for each eigenvalue has dimension equal to its multiplicity.
   4. Only if $A$ is symmetric