Exercise · ZW-DIAGONALIZATION-002 · ★☆☆ Basic
A \(2\times 2\) matrix has eigenvalues 1 and 4. Is it diagonalizable?
Choose one option.
Show answer
- A. Yes: distinct eigenvalues give two independent eigenvectors
Explanation
Eigenvectors of distinct eigenvalues are linearly independent; two of them form a basis of \(\mathbb{R}^2\).
Review the concept: Diagonalization
A square matrix \(A\) is diagonalizable if it is similar to a diagonal matrix \(D\), that is, if it can be written as an invertible matrix \(P\) times \(D\) times the inverse of \(P\).
\[A=PDP^{-1}\]
More practice problems: diagonalization
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- ZW-DIAGONALIZATION-006 A diagonalizable matrix has trace 7 and determinant 10. Which diagonal matrix is similar to it?
- ZW-DIAGONALIZATION-007 With A=[4 1; 2 3], A=PDP^-1 and D=diag(5,2), what is the diagonal matrix of A^5?
- ZW-DIAGONALIZATION-008 Let [1 1 0; 0 1 0; 0 0 2], with eigenvalues 1 (double) and 2. Is it diagonalizable?
- ZW-DIAGONALIZATION-009 A 3× 3 matrix has eigenvalues 2 (double) and 5 (simple). Which condition is equivalent to being dia…