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Is matrix$\left[\begin{array}{cc}0& 1\\ 5& 3\end{array}\right]$similar to $\left[\begin{array}{cc}1& 2\\ 4& 3\end{array}\right]$?

Find a $\mathbf{2}\mathbf{脳}\mathbf{2}$matrix A such that $\left[\begin{array}{c}3\\ 1\end{array}\right]$and $\left[\begin{array}{c}1\\ 2\end{array}\right]$ are eigenvectors of A , with eigenvalues 5 and 10 , respectively.

For an arbitrary positive integer n, give a $\mathbf{2}\mathbf{n}\mathbf{脳}\mathbf{2}\mathbf{n}$ matrix A without real eigenvalues.

consider an eigenvalue ${\mathit{位}}_{\mathbf{0}}$of an$\mathbf{n}\mathbf{脳}\mathbf{n}$matrix A. we are told that the algebraic multiplicity of exceeds 1.Show that$\mathbf{f}\mathbf{\text{'}}\left({位}_0\right)\mathbf{=}\mathbf{0}$(i.e.., the derivative of the characteristic polynomial of A vanishes are${\mathbf{位}}_{\mathbf{0}}$).

Consider the matrix

$\mathit{A}\mathbf{=}\left[\begin{array}{cc}3& 4\\ 4& -3\end{array}\right]$

a. Use the geometric interpretation of this transformation as a reflection combined with scaling to find the eigenvaluesA.

b. Find an eigen basis for A.

c. Diagonalize A .

Consider a symmetric nxnmatrix A.

a. Show that if$\stackrel{\mathbf{鈫�}}{\mathbf{v}}$and$\stackrel{\mathbf{鈫�}}{\mathit{w}}$are two vectors in${\mathbf{鈩�}}^{\mathbf{n}}$, then

$\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{A}\stackrel{\mathbf{鈫�}}{\mathbf{v}}\mathbf{.}\stackrel{\mathbf{鈫�}}{\mathbf{w}}\mathbf{=}\stackrel{\mathbf{鈫�}}{\mathbf{v}}\mathbf{.}\mathbf{A}\stackrel{\mathbf{鈫�}}{\mathbf{w}}\mathbf{.}$

b. Show that if$\stackrel{\mathbf{鈫�}}{\mathbf{v}}$and$\stackrel{\mathbf{鈫�}}{\mathbf{w}}$are two eigenvectors of A, with distinct eigenvalues, then$\stackrel{\mathbf{鈫�}}{\mathbf{w}}$is orthogonal to$\stackrel{\mathbf{鈫�}}{\mathbf{v}}$.

We are told that $\left[\begin{array}{c}1\\ -1\\ -1\end{array}\right]$is an eigenvector of the matrix$\left[\begin{array}{ccc}4& 1& 1\\ -5& 0& -3\\ -1& -1& 2\end{array}\right]$ what is the associated eigenvalue?

if A is a $\mathbf{2}\mathbf{脳}\mathbf{2}$matrix with t r A = 5and det A = - 14what are the eigenvalues of A?

If a 2 x 2matrix R represents a reflection about a line L, then R must be diagonalizable.

Consider a rotation$\mathbf{T}\mathbf{}\left(\stackrel{鈫�}{x}\right)\mathbf{=}\mathbf{A}\stackrel{\mathbf{鈫�}}{\mathbf{x}}\mathbf{}\mathbf{in}\mathbf{}{\mathbf{鈩�}}^{\mathbf{3}}$in. (That is, A is an orthogonal 3x3matrix with determinant 1.) Show that T has a non-zero fixed point [i.e., a vector$\mathbf{T}\mathbf{}\mathbf{\left(}\stackrel{\mathbf{鈫�}}{\mathbf{x}}\mathbf{\right)}\mathbf{=}\mathbf{A}\stackrel{\mathbf{鈫�}}{\mathbf{x}}\mathbf{}\mathbf{in}\mathbf{}{\mathbf{鈩�}}^{\mathbf{3}}$with$\mathbf{T}\mathbf{}\left(\stackrel{鈫�}{v}\right)\mathbf{=}\stackrel{\mathbf{鈫�}}{\mathbf{v}}$]. This result is known as Euler鈥檚 theorem, after the great Swiss mathematician Leonhard Euler (1707鈥�1783). Hint: Consider the characteristic polynomialrole="math" localid="1659595800447" ${\mathbf{f}}_{\mathbf{A}}\mathbf{\left(}\mathbf{位}\mathbf{\right)}$. Pay attention to the intercepts with both axes. Use Theorem 7.1.4.