Tagged: eigenspace
by Yu · Published 12/18/2017
Find Eigenvalues, Eigenvectors, and Diagonalize the 2 by 2 Matrix
Consider the matrix $A=\begin{bmatrix}
a & -b\\
b& a
\end{bmatrix}$, where $a$ and $b$ are real numbers and $b\neq 0$.
(a) Find all eigenvalues of $A$.
(b) For each eigenvalue of $A$, determine the eigenspace $E_{\lambda}$.
(c) Diagonalize the matrix $A$ by finding a nonsingular matrix $S$ and a diagonal matrix $D$ such that $S^{-1}AS=D$.
If Matrices Commute $AB=BA$, then They Share a Common Eigenvector
Let $A$ and $B$ be $n\times n$ matrices and assume that they commute: $AB=BA$.
Then prove that the matrices $A$ and $B$ share at least one common eigenvector.
Diagonalize the $2\times 2$ Hermitian Matrix by a Unitary Matrix
Consider the Hermitian matrix
\[A=\begin{bmatrix}
1 & i\\
-i& 1
\end{bmatrix}.\]
(a) Find the eigenvalues of $A$.
(b) For each eigenvalue of $A$, find the eigenvectors.
(c) Diagonalize the Hermitian matrix $A$ by a unitary matrix. Namely, find a diagonal matrix $D$ and a unitary matrix $U$ such that $U^{-1}AU=D$.
Unit Vectors and Idempotent Matrices
A square matrix $A$ is called idempotent if $A^2=A$.
(a) Let $\mathbf{u}$ be a vector in $\R^n$ with length $1$.
Define the matrix $P$ to be $P=\mathbf{u}\mathbf{u}^{\trans}$.
Prove that $P$ is an idempotent matrix.
(b) Suppose that $\mathbf{u}$ and $\mathbf{v}$ be unit vectors in $\R^n$ such that $\mathbf{u}$ and $\mathbf{v}$ are orthogonal.
Let $Q=\mathbf{u}\mathbf{u}^{\trans}+\mathbf{v}\mathbf{v}^{\trans}$.
Prove that $Q$ is an idempotent matrix.
(c) Prove that each nonzero vector of the form $a\mathbf{u}+b\mathbf{v}$ for some $a, b\in \R$ is an eigenvector corresponding to the eigenvalue $1$ for the matrix $Q$ in part (b).
Find a Basis of the Eigenspace Corresponding to a Given Eigenvalue
1 & 2 & 1 \\
-1 &4 &1 \\
2 & -4 & 0
\end{bmatrix}.\] The matrix $A$ has an eigenvalue $2$.
Find a basis of the eigenspace $E_2$ corresponding to the eigenvalue $2$.
(The Ohio State University, Linear Algebra Final Exam Problem)
Determine the Values of $a$ such that the 2 by 2 Matrix is Diagonalizable
1-a & a\\
-a& 1+a
\end{bmatrix}\] be a $2\times 2$ matrix, where $a$ is a complex number.
Determine the values of $a$ such that the matrix $A$ is diagonalizable.
(Nagoya University, Linear Algebra Exam Problem)
Diagonalize the 3 by 3 Matrix if it is Diagonalizable
Determine whether the matrix
0 & 0 & 2
\end{bmatrix}\] is diagonalizable.
If it is diagonalizable, then find the invertible matrix $S$ and a diagonal matrix $D$ such that $S^{-1}AS=D$.
Click here if solved 116
Idempotent Matrices are Diagonalizable
Let $A$ be an $n\times n$ idempotent matrix, that is, $A^2=A$. Then prove that $A$ is diagonalizable.
Find All the Eigenvalues and Eigenvectors of the 6 by 6 Matrix
Find all the eigenvalues and eigenvectors of the matrix
10001 & 3 & 5 & 7 &9 & 11 \\
1 & 10003 & 5 & 7 & 9 & 11 \\
1 & 3 & 10005 & 7 & 9 & 11 \\
1 & 3 & 5 & 10007 & 9 & 11 \\
1 &3 & 5 & 7 & 10009 & 11 \\
1 &3 & 5 & 7 & 9 & 10011
(MIT, Linear Algebra Homework Problem)
Quiz 13 (Part 2) Find Eigenvalues and Eigenvectors of a Special Matrix
Find all eigenvalues of the matrix
0 & i & i & i \\
i &0 & i & i \\
i & i & 0 & i \\
i & i & i & 0
\end{bmatrix},\] where $i=\sqrt{-1}$. For each eigenvalue of $A$, determine its algebraic multiplicity and geometric multiplicity.
Quiz 13 (Part 1) Diagonalize a Matrix
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 $S$ and a diagonal matrix $D$ such that $S^{-1}AS=D$.
Eigenvalues and Eigenvectors of Matrix Whose Diagonal Entries are 3 and 9 Elsewhere
3 & 9 & 9 & 9 \\
9 &3 & 9 & 9 \\
9 & 9 & 9 & 3
(Harvard University, Linear Algebra Final Exam Problem)
Eigenvalues and Algebraic/Geometric Multiplicities of Matrix $A+cI$
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?
Idempotent (Projective) Matrices are Diagonalizable
Let $A$ be an $n\times n$ idempotent complex matrix.
Then prove that $A$ is diagonalizable.
Quiz 12. Find Eigenvalues and their Algebraic and Geometric Multiplicities
(a) Let
\end{bmatrix}.\] Find the eigenvalues of the matrix $A$. Also give the algebraic multiplicity of each eigenvalue.
(b) Let
\end{bmatrix}.\] One of the eigenvalues of the matrix $A$ is $\lambda=0$. Find the geometric multiplicity of the eigenvalue $\lambda=0$.
Given Graphs of Characteristic Polynomial of Diagonalizable Matrices, Determine the Rank of Matrices
Let $A, B, C$ are $2\times 2$ diagonalizable matrices.
The graphs of characteristic polynomials of $A, B, C$ are shown below. The red graph is for $A$, the blue one for $B$, and the green one for $C$.
From this information, determine the rank of the matrices $A, B,$ and $C$.
Graphs of characteristic polynomials
Click here if solved 7
In this post, we explain how to diagonalize a matrix if it is diagonalizable.
As an example, we solve the following problem.
Diagonalize the matrix
3 &-2 &-3 \\
-1 & 1 & 2
\end{bmatrix}\] by finding a nonsingular matrix $S$ and a diagonal matrix $D$ such that $S^{-1}AS=D$.
(Update 10/15/2017. A new example problem was added.)
Maximize the Dimension of the Null Space of $A-aI$
\[ A=\begin{bmatrix}
5 & 2 & -1 \\
2 &2 &2 \\
Pick your favorite number $a$. Find the dimension of the null space of the matrix $A-aI$, where $I$ is the $3\times 3$ identity matrix.
Your score of this problem is equal to that dimension times five.
(The Ohio State University Linear Algebra Practice Problem)
Given All Eigenvalues and Eigenspaces, Compute a Matrix Product
Let $C$ be a $4 \times 4$ matrix with all eigenvalues $\lambda=2, -1$ and eigensapces
\[E_2=\Span\left \{\quad \begin{bmatrix}
1 \\
\end{bmatrix} \quad\right \} \text{ and } E_{-1}=\Span\left \{ \quad\begin{bmatrix}
\end{bmatrix},\quad \begin{bmatrix}
\end{bmatrix} \quad\right\}.\]
Calculate $C^4 \mathbf{u}$ for $\mathbf{u}=\begin{bmatrix}
\end{bmatrix}$ if possible. Explain why if it is not possible!
(The Ohio State University Linear Algebra Exam Problem)
Determine Eigenvalues, Eigenvectors, Diagonalizable From a Partial Information of a Matrix
Suppose the following information is known about a $3\times 3$ matrix $A$.
\[A\begin{bmatrix}
\end{bmatrix}=6\begin{bmatrix}
\end{bmatrix},
\quad
A\begin{bmatrix}
-1 \\
\end{bmatrix}, \quad
(b) Find the corresponding eigenspaces.
(c) In each of the following questions, you must give a correct reason (based on the theory of eigenvalues and eigenvectors) to get full credit.
Is $A$ a diagonalizable matrix?
Is $A$ an invertible matrix?
Is $A$ an idempotent matrix?
(Johns Hopkins University Linear Algebra Exam)
Find the Formula for the Power of a Matrix Using Linear Recurrence Relation
Are the Trigonometric Functions $\sin^2(x)$ and $\cos^2(x)$ Linearly Independent?
An Example of a Matrix that Cannot Be a Commutator
Matrix Representation of a Linear Transformation of Subspace of Sequences Satisfying Recurrence Relation
Matrix Representation of a Linear Transformation of the Vector Space $R^2$ to $R^2$