Category: Linear Algebra

Quiz 13 (Part 1) Diagonalize a Matrix

Problem 385

Let
\[A=\begin{bmatrix}
2 & -1 & -1 \\
-1 &2 &-1 \\
-1 & -1 & 2
\end{bmatrix}.\] Determine whether the matrix $A$ is diagonalizable. If it is diagonalizable, then diagonalize $A$.
That is, find a nonsingular matrix $A$ and a diagonal matrix $D$ such that $S^{-1}AS=D$.

 

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Determine Dimensions of Eigenspaces From Characteristic Polynomial of Diagonalizable Matrix

Problem 384

Let $A$ be an $n\times n$ matrix with the characteristic polynomial
\[p(t)=t^3(t-1)^2(t-2)^5(t+2)^4.\] Assume that the matrix $A$ is diagonalizable.

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

(b) Find the dimension of the eigenspace $E_2$ corresponding to the eigenvalue $\lambda=2$.

(c) Find the nullity of $A$.

(The Ohio State University, Linear Algebra final exam problem)
 

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Common Eigenvector of Two Matrices $A, B$ is Eigenvector of $A+B$ and $AB$.

Problem 382

Let $\lambda$ be an eigenvalue of $n\times n$ matrices $A$ and $B$ corresponding to the same eigenvector $\mathbf{x}$.

(a) Show that $2\lambda$ is an eigenvalue of $A+B$ corresponding to $\mathbf{x}$.

(b) Show that $\lambda^2$ is an eigenvalue of $AB$ corresponding to $\mathbf{x}$.

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

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Prove that the Length $\|A^n\mathbf{v}\|$ is As Small As We Like.

Problem 381

Consider the matrix
\[A=\begin{bmatrix}
3/2 & 2\\
-1& -3/2
\end{bmatrix} \in M_{2\times 2}(\R).\]

(a) Find the eigenvalues and corresponding eigenvectors of $A$.

(b) Show that for $\mathbf{v}=\begin{bmatrix}
1 \\
0
\end{bmatrix}\in \R^2$, we can choose $n$ large enough so that the length $\|A^n\mathbf{v}\|$ is as small as we like.

(University of California, Berkeley, Linear Algebra Final Exam Problem)
 

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Eigenvalues and Algebraic/Geometric Multiplicities of Matrix $A+cI$

Problem 378

Let $A$ be an $n \times n$ matrix and let $c$ be a complex number.

(a) For each eigenvalue $\lambda$ of $A$, prove that $\lambda+c$ is an eigenvalue of the matrix $A+cI$, where $I$ is the identity matrix. What can you say about the eigenvectors corresponding to $\lambda+c$?

(b) Prove that the algebraic multiplicity of the eigenvalue $\lambda$ of $A$ is the same as the algebraic multiplicity of the eigenvalue $\lambda+c$ of $A+cI$ are equal.

(c) How about geometric multiplicities?

 

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Quiz 12. Find Eigenvalues and their Algebraic and Geometric Multiplicities

Problem 376

(a) Let
\[A=\begin{bmatrix}
0 & 0 & 0 & 0 \\
1 &1 & 1 & 1 \\
0 & 0 & 0 & 0 \\
1 & 1 & 1 & 1
\end{bmatrix}.\] Find the eigenvalues of the matrix $A$. Also give the algebraic multiplicity of each eigenvalue.

(b) Let
\[A=\begin{bmatrix}
0 & 0 & 0 & 0 \\
1 &1 & 1 & 1 \\
0 & 0 & 0 & 0 \\
1 & 1 & 1 & 1
\end{bmatrix}.\] One of the eigenvalues of the matrix $A$ is $\lambda=0$. Find the geometric multiplicity of the eigenvalue $\lambda=0$.

 

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Powers of a Matrix Cannot be a Basis of the Vector Space of Matrices

Problem 375

Let $n>1$ be a positive integer. Let $V=M_{n\times n}(\C)$ be the vector space over the complex numbers $\C$ consisting of all complex $n\times n$ matrices. The dimension of $V$ is $n^2$.
Let $A \in V$ and consider the set
\[S_A=\{I=A^0, A, A^2, \dots, A^{n^2-1}\}\] of $n^2$ elements.
Prove that the set $S_A$ cannot be a basis of the vector space $V$ for any $A\in V$.

 

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Determinant of a General Circulant Matrix

Problem 374

Let \[A=\begin{bmatrix}
a_0 & a_1 & \dots & a_{n-2} &a_{n-1} \\
a_{n-1} & a_0 & \dots & a_{n-3} & a_{n-2} \\
a_{n-2} & a_{n-1} & \dots & a_{n-4} & a_{n-3} \\
\vdots & \vdots & \dots & \vdots & \vdots \\
a_{2} & a_3 & \dots & a_{0} & a_{1}\\
a_{1} & a_2 & \dots & a_{n-1} & a_{0}
\end{bmatrix}\] be a complex $n \times n$ matrix.
Such a matrix is called circulant matrix.
Then prove that the determinant of the circulant matrix $A$ is given by
\[\det(A)=\prod_{k=0}^{n-1}(a_0+a_1\zeta^k+a_2 \zeta^{2k}+\cdots+a_{n-1}\zeta^{k(n-1)}),\] where $\zeta=e^{2 \pi i/n}$ is a primitive $n$-th root of unity.

 

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Compute Power of Matrix If Eigenvalues and Eigenvectors Are Given

Problem 373

Let $A$ be a $3\times 3$ matrix. Suppose that $A$ has eigenvalues $2$ and $-1$, and suppose that $\mathbf{u}$ and $\mathbf{v}$ are eigenvectors corresponding to $2$ and $-1$, respectively, where
\[\mathbf{u}=\begin{bmatrix}
1 \\
0 \\
-1
\end{bmatrix} \text{ and } \mathbf{v}=\begin{bmatrix}
2 \\
1 \\
0
\end{bmatrix}.\] Then compute $A^5\mathbf{w}$, where
\[\mathbf{w}=\begin{bmatrix}
7 \\
2 \\
-3
\end{bmatrix}.\]

 

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Find Matrix Representation of Linear Transformation From $\R^2$ to $\R^2$

Problem 370

Let $T: \R^2 \to \R^2$ be a linear transformation such that
\[T\left(\, \begin{bmatrix}
1 \\
1
\end{bmatrix} \,\right)=\begin{bmatrix}
4 \\
1
\end{bmatrix}, T\left(\, \begin{bmatrix}
0 \\
1
\end{bmatrix} \,\right)=\begin{bmatrix}
3 \\
2
\end{bmatrix}.\] Then find the matrix $A$ such that $T(\mathbf{x})=A\mathbf{x}$ for every $\mathbf{x}\in \R^2$, and find the rank and nullity of $T$.

(The Ohio State University, Linear Algebra Exam Problem)
 

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Rank and Nullity of Linear Transformation From $\R^3$ to $\R^2$

Problem 369

Let $T:\R^3 \to \R^2$ be a linear transformation such that
\[ T(\mathbf{e}_1)=\begin{bmatrix}
1 \\
0
\end{bmatrix}, T(\mathbf{e}_2)=\begin{bmatrix}
0 \\
1
\end{bmatrix}, T(\mathbf{e}_3)=\begin{bmatrix}
1 \\
0
\end{bmatrix},\] where $\mathbf{e}_1, \mathbf{e}_2, \mathbf{e}_3$ are the standard basis of $\R^3$.
Then find the rank and the nullity of $T$.

(The Ohio State University, Linear Algebra Exam Problem)
 

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Determine a Value of Linear Transformation From $\R^3$ to $\R^2$

Problem 368

Let $T$ be a linear transformation from $\R^3$ to $\R^2$ such that
\[ T\left(\, \begin{bmatrix}
0 \\
1 \\
0
\end{bmatrix}\,\right) =\begin{bmatrix}
1 \\
2
\end{bmatrix} \text{ and }T\left(\, \begin{bmatrix}
0 \\
1 \\
1
\end{bmatrix}\,\right)=\begin{bmatrix}
0 \\
1
\end{bmatrix}. \] Then find $T\left(\, \begin{bmatrix}
0 \\
1 \\
2
\end{bmatrix} \,\right)$.

 
(The Ohio State University, Linear Algebra Exam Problem)

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Basis of Span in Vector Space of Polynomials of Degree 2 or Less

Problem 367

Let $P_2$ be the vector space of all polynomials of degree $2$ or less with real coefficients.
Let
\[S=\{1+x+2x^2, \quad x+2x^2, \quad -1, \quad x^2\}\] be the set of four vectors in $P_2$.

Then find a basis of the subspace $\Span(S)$ among the vectors in $S$.

(Linear Algebra Exam Problem, the Ohio State University)
 

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