Tagged: eigenvalue

Compute $A^{10}\mathbf{v}$ Using Eigenvalues and Eigenvectors of the Matrix $A$

Problem 485

Let
\[A=\begin{bmatrix}
1 & -14 & 4 \\
-1 &6 &-2 \\
-2 & 24 & -7
\end{bmatrix} \quad \text{ and }\quad \mathbf{v}=\begin{bmatrix}
4 \\
-1 \\
-7
\end{bmatrix}.\] Find $A^{10}\mathbf{v}$.

You may use the following information without proving it.
The eigenvalues of $A$ are $-1, 0, 1$. The eigenspaces are given by
\[E_{-1}=\Span\left\{\, \begin{bmatrix}
3 \\
-1 \\
-5
\end{bmatrix} \,\right\}, \quad E_{0}=\Span\left\{\, \begin{bmatrix}
-2 \\
1 \\
4
\end{bmatrix} \,\right\}, \quad E_{1}=\Span\left\{\, \begin{bmatrix}
-4 \\
2 \\
7
\end{bmatrix} \,\right\}.\]

(The Ohio State University, Linear Algebra Final Exam Problem)

 
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A Linear Transformation Preserves Exactly Two Lines If and Only If There are Two Real Non-Zero Eigenvalues

Problem 472

Let $T:\R^2 \to \R^2$ be a linear transformation and let $A$ be the matrix representation of $T$ with respect to the standard basis of $\R^2$.

Prove that the following two statements are equivalent.

(a) There are exactly two distinct lines $L_1, L_2$ in $\R^2$ passing through the origin that are mapped onto themselves:
\[T(L_1)=L_1 \text{ and } T(L_2)=L_2.\]

(b) The matrix $A$ has two distinct nonzero real eigenvalues.

 
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Use the Cayley-Hamilton Theorem to Compute the Power $A^{100}$

Problem 471

Let $A$ be a $3\times 3$ real orthogonal matrix with $\det(A)=1$.

(a) If $\frac{-1+\sqrt{3}i}{2}$ is one of the eigenvalues of $A$, then find the all the eigenvalues of $A$.

(b) Let
\[A^{100}=aA^2+bA+cI,\] where $I$ is the $3\times 3$ identity matrix.
Using the Cayley-Hamilton theorem, determine $a, b, c$.

(Kyushu University, Linear Algebra Exam Problem)
 
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Diagonalize a 2 by 2 Matrix $A$ and Calculate the Power $A^{100}$

Problem 466

Let
\[A=\begin{bmatrix}
1 & 2\\
4& 3
\end{bmatrix}.\]

(a) Find eigenvalues of the matrix $A$.

(b) Find eigenvectors for each eigenvalue of $A$.

(c) Diagonalize the matrix $A$. That is, find an invertible matrix $S$ and a diagonal matrix $D$ such that $S^{-1}AS=D$.

(d) Diagonalize the matrix $A^3-5A^2+3A+I$, where $I$ is the $2\times 2$ identity matrix.

(e) Calculate $A^{100}$. (You do not have to compute $5^{100}$.)

(f) Calculate
\[(A^3-5A^2+3A+I)^{100}.\] Let $w=2^{100}$. Express the solution in terms of $w$.

 
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True of False Problems on Determinants and Invertible Matrices

Problem 438

Determine whether each of the following statements is True or False.

(a) If $A$ and $B$ are $n \times n$ matrices, and $P$ is an invertible $n \times n$ matrix such that $A=PBP^{-1}$, then $\det(A)=\det(B)$.

(b) If the characteristic polynomial of an $n \times n$ matrix $A$ is
\[p(\lambda)=(\lambda-1)^n+2,\] then $A$ is invertible.

(c) If $A^2$ is an invertible $n\times n$ matrix, then $A^3$ is also invertible.

(d) If $A$ is a $3\times 3$ matrix such that $\det(A)=7$, then $\det(2A^{\trans}A^{-1})=2$.

(e) If $\mathbf{v}$ is an eigenvector of an $n \times n$ matrix $A$ with corresponding eigenvalue $\lambda_1$, and if $\mathbf{w}$ is an eigenvector of $A$ with corresponding eigenvalue $\lambda_2$, then $\mathbf{v}+\mathbf{w}$ is an eigenvector of $A$ with corresponding eigenvalue $\lambda_1+\lambda_2$.

(Stanford University, Linear Algebra Exam Problem)
 
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If Two Matrices Have the Same Eigenvalues with Linearly Independent Eigenvectors, then They Are Equal

Problem 424

Let $A$ and $B$ be $n\times n$ matrices.
Suppose that $A$ and $B$ have the same eigenvalues $\lambda_1, \dots, \lambda_n$ with the same corresponding eigenvectors $\mathbf{x}_1, \dots, \mathbf{x}_n$.
Prove that if the eigenvectors $\mathbf{x}_1, \dots, \mathbf{x}_n$ are linearly independent, then $A=B$.

 
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