Exercise · ZW-DIAGONALIZATION-001 · ★☆☆ Basic

Let \(\begin{pmatrix}4&1\\2&3\end{pmatrix}\) with eigenvalues 5 and 2. With \(D=\operatorname{diag}(5,2)\), which matrix \(P\) satisfies \(P^{-1}AP=D\)?

Choose one option.

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  • C. \(\begin{pmatrix}1&1\\1&-2\end{pmatrix}\)

Explanation

The columns of \(P\) are eigenvectors in the order of \(D\): \((1,1)\) for 5 and \((1,-2)\) for 2. Swapping columns would swap the diagonal order.

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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

ZW-DIAGONALIZATION-002 →