Eigenvalues and their Algebraic Multiplicities of a Matrix with a Variable

Problems and Solutions of Eigenvalue, Eigenvector in Linear Algebra

Problem 206

Determine all eigenvalues and their algebraic multiplicities of the matrix
\[A=\begin{bmatrix}
1 & a & 1 \\
a &1 &a \\
1 & a & 1
\end{bmatrix},\] where $a$ is a real number.

 
LoadingAdd to solve later

Sponsored Links

Proof.

To find eigenvalues we first compute the characteristic polynomial of the matrix $A$ as follows.
\begin{align*}
\det(A-tI)&=\begin{vmatrix}
1-t & a & 1 \\
a &1-t &a \\
1 & a & 1-t
\end{vmatrix}\\
&=(1-t)\begin{vmatrix}
1-t & a\\
a& 1-t
\end{vmatrix}-a\begin{vmatrix}
a & a\\
1& 1-t
\end{vmatrix}+\begin{vmatrix}
a & 1-t\\
1& a
\end{vmatrix}
\end{align*}
We used the first row cofactor expansion in the second equality.


After we compute three $2 \times 2$ determinants and simply, we obtain
\[\det(A-tI)=-t(t^2-3t+2-2a^2).\] The eigenvalues of $A$ are roots of this characteristic polynomial. Thus, eigenvalues are
\[0, \quad \frac{3\pm \sqrt{1+8a^2}}{2}\] by the quadratic formula.


Now the only possible way to obtain a multiplicity $2$ eigenvalue is when
\[\frac{3- \sqrt{1+8a^2}}{2}=0\] and it is straightforward to check that this happens if and only if $a=1$.

Therefore, when $a=1$ eigenvalues of $A$ are $0$ with algebraic multiplicity $2$ and $3$ with algebraic multiplicity $1$.

When $a \neq 1$, eigenvalues are
\[0, \quad \frac{3\pm \sqrt{1+8a^2}}{2}\] and each of them has algebraic multiplicity $1$.


LoadingAdd to solve later

Sponsored Links

More from my site

  • Maximize the Dimension of the Null Space of $A-aI$Maximize the Dimension of the Null Space of $A-aI$ Let \[ A=\begin{bmatrix} 5 & 2 & -1 \\ 2 &2 &2 \\ -1 & 2 & 5 \end{bmatrix}.\] 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 […]
  • Determine a Matrix From Its EigenvalueDetermine a Matrix From Its Eigenvalue Let \[A=\begin{bmatrix} a & -1\\ 1& 4 \end{bmatrix}\] be a $2\times 2$ matrix, where $a$ is some real number. Suppose that the matrix $A$ has an eigenvalue $3$. (a) Determine the value of $a$. (b) Does the matrix $A$ have eigenvalues other than […]
  • Find All the Eigenvalues of Power of Matrix and Inverse MatrixFind All the Eigenvalues of Power of Matrix and Inverse Matrix Let \[A=\begin{bmatrix} 3 & -12 & 4 \\ -1 &0 &-2 \\ -1 & 5 & -1 \end{bmatrix}.\] Then find all eigenvalues of $A^5$. If $A$ is invertible, then find all the eigenvalues of $A^{-1}$.   Proof. We first determine all the eigenvalues of the matrix […]
  • Characteristic Polynomial, Eigenvalues, Diagonalization Problem (Princeton University Exam)Characteristic Polynomial, Eigenvalues, Diagonalization Problem (Princeton University Exam) Let \[\begin{bmatrix} 0 & 0 & 1 \\ 1 &0 &0 \\ 0 & 1 & 0 \end{bmatrix}.\] (a) Find the characteristic polynomial and all the eigenvalues (real and complex) of $A$. Is $A$ diagonalizable over the complex numbers? (b) Calculate $A^{2009}$. (Princeton University, […]
  • Rotation Matrix in Space and its Determinant and EigenvaluesRotation Matrix in Space and its Determinant and Eigenvalues For a real number $0\leq \theta \leq \pi$, we define the real $3\times 3$ matrix $A$ by \[A=\begin{bmatrix} \cos\theta & -\sin\theta & 0 \\ \sin\theta &\cos\theta &0 \\ 0 & 0 & 1 \end{bmatrix}.\] (a) Find the determinant of the matrix $A$. (b) Show that $A$ is an […]
  • How to Use the Cayley-Hamilton Theorem to Find the Inverse MatrixHow to Use the Cayley-Hamilton Theorem to Find the Inverse Matrix Find the inverse matrix of the $3\times 3$ matrix \[A=\begin{bmatrix} 7 & 2 & -2 \\ -6 &-1 &2 \\ 6 & 2 & -1 \end{bmatrix}\] using the Cayley-Hamilton theorem.   Solution. To apply the Cayley-Hamilton theorem, we first determine the characteristic […]
  • A Matrix Having One Positive Eigenvalue and One Negative EigenvalueA Matrix Having One Positive Eigenvalue and One Negative Eigenvalue Prove that the matrix \[A=\begin{bmatrix} 1 & 1.00001 & 1 \\ 1.00001 &1 &1.00001 \\ 1 & 1.00001 & 1 \end{bmatrix}\] has one positive eigenvalue and one negative eigenvalue. (University of California, Berkeley Qualifying Exam Problem)   Solution. Let us put […]
  • Find the Inverse Matrix Using the Cayley-Hamilton TheoremFind the Inverse Matrix Using the Cayley-Hamilton Theorem Find the inverse matrix of the matrix \[A=\begin{bmatrix} 1 & 1 & 2 \\ 9 &2 &0 \\ 5 & 0 & 3 \end{bmatrix}\] using the Cayley–Hamilton theorem.   Solution. To use the Cayley-Hamilton theorem, we first compute the characteristic polynomial $p(t)$ of […]

You may also like...

Leave a Reply

Your email address will not be published. Required fields are marked *

This site uses Akismet to reduce spam. Learn how your comment data is processed.

More in Linear Algebra
Ohio State University exam problems and solutions in mathematics
Eigenvalues of a Hermitian Matrix are Real Numbers

Show that eigenvalues of a Hermitian matrix $A$ are real numbers. (The Ohio State University Linear Algebra Exam Problem)  

Close