A Recursive Relationship for a Power of a Matrix

Problem 685

Suppose that the $2 \times 2$ matrix $A$ has eigenvalues $4$ and $-2$. For each integer $n \geq 1$, there are real numbers $b_n , c_n$ which satisfy the relation
$A^{n} = b_n A + c_n I ,$ where $I$ is the identity matrix.

Find $b_n$ and $c_n$ for $2 \leq n \leq 5$, and then find a recursive relationship to find $b_n, c_n$ for every $n \geq 1$.

Solution.

Because the eigenvalues of $A$ are $4$ and $-2$, its characteristic polynomial must be
$p(\lambda) = (\lambda – 4) ( \lambda + 2) = \lambda^2 – 2 \lambda – 8 .$

The Cayley-Hamilton Theorem tells us that $p(A) = 0$, the zero matrix. Rearranging terms, we find that
$A^2 = 2 A + 8 I .$ So $b_2=2$ and $c_2=8$.

With this relationship, we can reduce the higher powers of $A$:
\begin{align*}
A^3 &= A^2 A= (2A + 8 I)A = 2A^2 + 8 A\\
& = 2(2A+8I)+8A= 12 A + 16 I
\end{align*}
\begin{align*}
A^4 &= A^3 A=(12 A + 16 I)A = 12A^2+16A\\
&=12(2A+8I)+16A = 40 A + 96 I
\end{align*}
\begin{align*}
A^5 &=A^4A= (40 A + 96 I)A = 40 A^2 + 96 A \\
&=40(2A+8I)+96A = 176 A + 320 I
\end{align*}
Hence, we have $b_3=12, c_3=16, b_4=40, c_4=96, b_5=176$, and $c_5=320$.

To find the recursive relationship, suppose we know that $A^n = b_n A + c_n I$. Then
\begin{align*}
A^{n+1} &= A^{n} A = ( b_n A + c_n I ) A = b_n A^2 + c_n A\\
&=b_n(2A+8I)+c_n A = (2 b_n + c_n) A + 8 b_n I.
\end{align*}
This gives the recursive relationships
$b_{n+1} = 2 b_n + c_n , \qquad c_{n+1} = 8 b_n .$

Using the work above, you can quickly verify this for $1 \leq n \leq 4$.

More from my site

• Is the Derivative Linear Transformation Diagonalizable? Let $\mathrm{P}_2$ denote the vector space of polynomials of degree $2$ or less, and let $T : \mathrm{P}_2 \rightarrow \mathrm{P}_2$ be the derivative linear transformation, defined by $T( ax^2 + bx + c ) = 2ax + b .$ Is $T$ diagonalizable? If so, find a diagonal matrix which […]
• Commuting Matrices $AB=BA$ such that $A-B$ is Nilpotent Have the Same Eigenvalues Let $A$ and $B$ be square matrices such that they commute each other: $AB=BA$. Assume that $A-B$ is a nilpotent matrix. Then prove that the eigenvalues of $A$ and $B$ are the same.   Proof. Let $N:=A-B$. By assumption, the matrix $N$ is nilpotent. This […]
• True or False: Eigenvalues of a Real Matrix Are Real Numbers Answer the following questions regarding eigenvalues of a real matrix. (a) True or False. If each entry of an $n \times n$ matrix $A$ is a real number, then the eigenvalues of $A$ are all real numbers. (b) Find the eigenvalues of the matrix $B=\begin{bmatrix} -2 & […] • An Example of a Matrix that Cannot Be a Commutator Let I be the 2\times 2 identity matrix. Then prove that -I cannot be a commutator [A, B]:=ABA^{-1}B^{-1} for any 2\times 2 matrices A and B with determinant 1. Proof. Assume that [A, B]=-I. Then ABA^{-1}B^{-1}=-I implies \[ABA^{-1}=-B. […] • Find the Formula for the Power of a Matrix Using Linear Recurrence Relation Suppose that A is 2\times 2 matrix that has eigenvalues -1 and 3. Then for each positive integer n find a_n and b_n such that \[A^{n+1}=a_nA+b_nI,$ where $I$ is the $2\times 2$ identity matrix.   Solution. Since $-1, 3$ are eigenvalues of the […]
• How to Calculate and Simplify a Matrix Polynomial Let $T=\begin{bmatrix} 1 & 0 & 2 \\ 0 &1 &1 \\ 0 & 0 & 2 \end{bmatrix}$. Calculate and simplify the expression $-T^3+4T^2+5T-2I,$ where $I$ is the $3\times 3$ identity matrix. (The Ohio State University Linear Algebra Exam) Hint. Use the […]
• The Formula for the Inverse Matrix of $I+A$ for a $2\times 2$ Singular Matrix $A$ Let $A$ be a singular $2\times 2$ matrix such that $\tr(A)\neq -1$ and let $I$ be the $2\times 2$ identity matrix. Then prove that the inverse matrix of the matrix $I+A$ is given by the following formula: $(I+A)^{-1}=I-\frac{1}{1+\tr(A)}A.$ Using the formula, calculate […]
• Find 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...

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

The Rotation Matrix is an Orthogonal Transformation

Let $\mathbb{R}^2$ be the vector space of size-2 column vectors. This vector space has an inner product defined by \$...

Close