Eigenvalues, eigenvectors and diagonalization

Determine the eigenvalues and eigenspaces of the matrix

$$A=\left(\begin{array}{cc}3& 1\\ 2& 2\end{array}\right).$$

Solution available to those who have unlocked the subject.

Determine $k\in \mathbb{R}$ so that $v=(1,1)$ is an eigenvector of the matrix

$$A=\left(\begin{array}{cc}1& k\\ 2& 3\end{array}\right),$$

and for that value compute the eigenvalues and the eigenspaces of $A$.

Solution available to those who have unlocked the subject.

Determine the eigenvalues, their algebraic and geometric multiplicities and a basis of each eigenspace of the matrix

$$A=\left(\begin{array}{ccc}1& 2& 0\\ 0& 3& 0\\ -4& 4& 3\end{array}\right).$$

Solution available to those who have unlocked the subject.

Verify that $v=(1,1,1)$ is an eigenvector of the matrix

$$A=\left(\begin{array}{ccc}2& 1& -1\\ 0& 3& -1\\ 1& 1& 0\end{array}\right),$$

determine all the eigenvalues of $A$ and determine whether $A$ is diagonalizable.

Solution available to those who have unlocked the subject.

Find an invertible matrix $P$ and a diagonal matrix $D$ such that $P^{-1}AP=D$, where

$$A=\left(\begin{array}{cc}4& -2\\ 1& 1\end{array}\right).$$

Solution available to those who have unlocked the subject.

Diagonalize the matrix

$$A=\left(\begin{array}{ccc}1& 0& 2\\ 0& 2& 0\\ 0& 1& 3\end{array}\right),$$

that is, find an invertible $P$ and a diagonal $D$ such that $P^{-1}AP=D$.

Solution available to those who have unlocked the subject.

Determine whether the matrix

$$A=\left(\begin{array}{ccc}-1& 3& 0\\ 0& 2& 0\\ 3& -2& 2\end{array}\right)$$

is diagonalizable.

Solution available to those who have unlocked the subject.

Determine the real eigenvalues and the corresponding eigenspaces of the matrix

$$A=\left(\begin{array}{ccc}1& 0& 0\\ 0& 1& -1\\ 0& 1& 1\end{array}\right)$$

and determine whether $A$ is diagonalizable over $\mathbb{R}$.

Solution available to those who have unlocked the subject.

Determine for which $k\in \mathbb{R}$ the matrix

$$A=\left(\begin{array}{ccc}1& k& 0\\ 1& 1& 0\\ 0& 0& 2\end{array}\right)$$

is diagonalizable.

Solution available to those who have unlocked the subject.

Determine for which $k\in \mathbb{R}$ the matrix

$$A=\left(\begin{array}{ccc}2& k& 0\\ 0& 2& 0\\ 0& 0& k\end{array}\right)$$

is diagonalizable.

Solution available to those who have unlocked the subject.

Compute $A^n$ for every integer $n\ge 1$, where

$$A=\left(\begin{array}{cc}-4& 3\\ -6& 5\end{array}\right).$$

Solution available to those who have unlocked the subject.

Let $f:{\mathbb{R}}_2[x]\to {\mathbb{R}}_2[x]$ be the endomorphism defined by $f(p(x))=p(x+1)$. Determine the eigenvalues and eigenspaces of $f$ and determine whether $f$ is diagonalizable.

Solution available to those who have unlocked the subject.

Let $f:{\mathbb{R}}^3\to {\mathbb{R}}^3$ be the endomorphism such that

$$f(1,1,0)=(2,2,0),f(0,1,1)=(0,2,2),f(1,0,1)=(0,0,0).$$

Determine the eigenvalues and eigenspaces of $f$, determine whether $f$ is diagonalizable and write the matrix associated with $f$ with respect to the standard basis.

Solution available to those who have unlocked the subject.