Diagonalizability

Repeat Example 2.5 for the matrix from Example 2.2.

Diagonalize this matrix by following the steps of Example 2.5.

• Set up the matrix-vector equation described in Lemma 2.4 and rewrite it as a linear system.

• By considering solutions for that system, find two vectors to make a basis. (Consider separately the system in the $x=0$ and $x\neq 0$ cases. Also, recall that the zero vector cannot be a member of a basis.)

• Use that basis in the similarity diagram t…

Follow Example 2.5 to diagonalize this matrix.

• Set up the matrix-vector equation and rewrite it as a linear system.

• By considering solutions for that system, find two vectors to make a basis. (Consider separately the $x=0$ and $x\neq 0$ cases. Also, recall that the zero vector cannot be a member of a basis.)

• With that basis use the similarity diagram to get the diagonalization as the product of three m…

Diagonalize these upper triangular matrices.

• (a)  $\begin{pmatrix}-2&1\\ 0&2\end{pmatrix}$     (b)  $\begin{pmatrix}5&4\\ 0&1\end{pmatrix}$

If we try to diagonalize the matrix of Example 2.3

using the method of Example 2.5 then what goes wrong?

• Draw the similarity diagram with $N$.

• Set up the matrix-vector equation described in Lemma 2.4 and rewrite it as a linear system.

• By considering solutions for that system, find the trouble. (Consider separately the $x=0$ and $x\neq 0$ cases.)

What form do the powers of a diagonal matrix have?

Give two same-sized diagonal matrices that are not similar. Must any two different diagonal matrices come from different similarity classes?

Give a nonsingular diagonal matrix. Can a diagonal matrix ever be singular?

Show that the inverse of a diagonal matrix is the diagonal of the inverses, if no element on that diagonal is zero. What happens when a diagonal entry is zero?

The equation ending Example 2.5

is a bit jarring because for $P$ we must take the first matrix, which is shown as an inverse, and for $P^{-1}$ we take the inverse of the first matrix, so that the two $-1$ powers cancel and this matrix is shown without a superscript $-1$.

• Check that this nicer-appearing equation holds.

• Is the previous item a coincidence? Or can we always switch the $P$ and the $P^{-1}$?

Show that the $P$ used to diagonalize in Example 2.5 is not unique.

Find a formula for the powers of this matrix. Hint: see Exercise 6.

We can ask how diagonalization interacts with the matrix operations. Assume that $t,s\colon V\to V$ are each diagonalizable. Is $ct$ diagonalizable for all scalars $c$? What about $t+s$? $t\mathbin{\circ}s$?

Show that matrices of this form are not diagonalizable.

Show that each of these is diagonalizable.

• (a)  $\begin{pmatrix}1&2\\ 2&1\end{pmatrix}$     (b)  $\begin{pmatrix}x&y\\ y&z\end{pmatrix}\qquad\text{$x,y,z$ scalars}$