Suggested languages for you:

Americas

Europe

Q25SE

Expert-verified
Found in: Page 267

### Linear Algebra and its Applications

Book edition 5th
Author(s) David C. Lay, Steven R. Lay and Judi J. McDonald
Pages 483 pages
ISBN 978-03219822384

# (M) Use a matrix program to diagonalizeA = \left( {\begin{aligned}{*{20}{c}}{ - 3}&{ - 2}&0\\{14}&7&{ - 1}\\{ - 6}&{ - 3}&1\end{aligned}} \right)If possible. Use the eigenvalue command to create the diagonal matrix $$D$$. If the program has a command that produces eigenvectors, use it to create an invertible matrix $$P$$. Then compute $$AP - PD$$ and $$PD{P^{{\bf{ - 1}}}}$$. Discuss your results.

$$A$$ is not diagonalizable. Use command eig in MATLAB to get invertible matrix $$P$$ and diagonal matrix $$D$$ if possible.

See the step by step solution

## Step 1: Determine whether $$P$$ is invertible or not

First, we need to define a matrix $$A$$ in MATLAB and use command eig to get an invertible matrix $$P$$ and a diagonal matrix $$D$$.

\left( {\begin{aligned}{*{20}{c}}P&D\end{aligned}} \right) = {\rm{eig}}\left( A \right)

Therefore,

P = \left( {\begin{aligned}{*{20}{c}}{0.3244}&{ - 0.3244}&{0.3333}\\{ - 0.8111}&{0.8111}&{ - 0.6667}\\{0.4867}&{ - 0.4867}&{0.6667}\end{aligned}} \right)

And

D = \left( {\begin{aligned}{*{20}{c}}2&0&0\\0&2&0\\0&0&1\end{aligned}} \right)

As the matrix $$A$$ has two eigenvalues $$2$$ and $$1$$.

Since the first two columns of the matrix are linearly dependent that means the eigenvalue $$2$$ is dimensional that shows $$P$$ is not invertible.

Therefore $$A$$ is not diagonalizable.

## Step 2: Compute $$AP - PD$$ and $$PD{P^{{\bf{ - 1}}}}$$

Since the matrix $$P$$ is not invertible therefore $$PD{P^{ - 1}}$$ cannot be computed.

However, $$AP - PD$$ can be computed.

$$AP$$

$$PD$$

$$AP - PD$$

As $$AP - PD$$ is equal to zero matrix.

Since $$P$$ consists of eigenvectors of $$A$$ therefore columns of $$AP - PD$$ are eigenvectors of $$A$$ which are multiplied by its eigenvalues.

Also, $$D$$ consists of eigenvalues of $$A$$ and columns of $$PD$$ are eigenvectors of $$A$$ multiplied by its eigenvalues.

Thus, use command eig in MATLAB to get invertible matrix $$P$$ and diagonal matrix $$D$$ if possible.