Questions tagged [characteristic-polynomial]
The characteristic polynomial of a square matrix is a polynomial that is invariant under matrix similarity and has the eigenvalues as roots.
50
questions
0
votes
1answer
30 views
Fibonacci closed form via vector space of infinite sequences of real numbers and geometric sequences
This question is from Linear Algebra with Applications (5th edition) by Otto Bretscher. It is in section 4.1 (Introduction to vector spaces), question 60. Below is the question:
Consider the ...
1
vote
2answers
45 views
$ A^3 + B^2 = I_n $ and $A^5=A^2$, then $\det(A^2 + B^2 + I_n) \geq 0 $ and $\operatorname{rank}(I_n + AB^2) = \mathrm{rank}(I_n - AB^2) $
Let $A, B$ be square matrices of size $n$, $n \geq 2$, containing real entries.
$\DeclareMathOperator\rank{rank}$
If the following properties take place: $ A^3 + B^2 = I_n $ and $A^5=A^2$,
then $...
1
vote
1answer
14 views
Matrix representation and characterstic polynomial of linear transformation from the 0 vector space to a finitely-dimensional space
Relevant definition:
In this definition, what happens if $V = \{0\}$, the 0 vector space? The basis of the zero vector space is the empty set, so the definition would not make much sense. Do we just ...
0
votes
4answers
25 views
Characteristic polynomial in $\mathbb{C}^2$
Consider a finite-dimensional vector space $V$ over a field $\mathbb{F}$.Let $a,b\in\mathbb{F}$. State whether the following is true or not: If $\begin{pmatrix} b & a \\ 0 & b \end{pmatrix}$ ...
-1
votes
2answers
47 views
diagonalize the following n×n matrix. I am wondering if my solution for the characteristic polynomial is valid or if there is a better way to do it.
The following is the question:
The following is my answer.
1
vote
2answers
65 views
Diagonalizability of a matrix
Show that $$ A :=\begin{pmatrix} 1 & 0 & 0 \\ -2 & 1 & 2 \\ -2 & 0 & 3 \end{pmatrix}$$ is diagonalizable.
What I did:
First, I determined the characteristic polynomial $$\...
5
votes
2answers
86 views
$ A^2 - B^2 = I_{2n+1} \implies det(AB-BA)=0 $ where A,B are complex matrices of odd size
Let $A, B$ be square matrices (with complex entries) of size $2n+1$, where $n$ is a positive integer.
I need help proving the following: $$A^2 - B^2 = I_{2n+1} \implies det(AB-BA)=0 $$
I've tried ...
1
vote
3answers
97 views
Calculate matrix by using Cayley-Hamilton theorem
Calculate matrix $B = A^{10}-3A^9-A^2+4A$ using Cayley-Hamilton theorem on $A$.
$$A = \begin{pmatrix}
2 & 2 & 2 & 5 \\
-1 & -1 & -1 & -5 \\
-2 & -2 & -1 & 0 \\...
1
vote
1answer
58 views
Kernel of characteristic polynomial of linear transformation
Let $F$ be a Field and $V$ be a Vectorspace with $\dim(V)=n$. Let $f:V \to V$ be a linear transformation with the characteristic polynomial $pa(x)$.
It is linear factorized. Let $pa(x)=g(x)*h(x)$ ...
1
vote
1answer
48 views
Drawing conclusions from a characteristic polynomial
I was given a characteristic polynomial and was asked to draw some conclusions from it, but i need some help with figuring something out. My polynomial is this : $\lambda^4 -3\lambda^3 +\lambda^2 +3\...
3
votes
1answer
48 views
Characteristic polynomial of a matrix — algorithm
Is there a classical algorithm to compute the coefficients of the characteristic polynomial of a real matrix, for small matrix sizes (say up to $10\times10$)?
Is there a specialized version for ...
0
votes
3answers
57 views
Given the characteristic equation, how to find the determinant of a matrix
Take a look at this question:
Find $\det(A)$ given that A has $p(\lambda)$ as its characteristic polynomial.
$$
p(\lambda) = \lambda^3 - 2\lambda^2 + \lambda + 5
$$
My first step is to notice the ...
1
vote
4answers
86 views
Find the $n$-th power of a $3{\times}3$ matrix using the Cayley-Hamilton theorem.
I need to find $A^n$ of the matrix $A=\begin{pmatrix}
2&0 & 2\\
0& 2 & 1\\
0& 0 & 3
\end{pmatrix}$ using Cayley-Hamilton theorem.
I found the characteristic polynomial $...
1
vote
0answers
24 views
$c_{n-1}$-th factor of polynomial and the trace of a matrix using induction
I've been looking around stack exchange, but I've been having some issues with this question.
Let $A\in M_{n,n}(K)$, for some field $K$. We write $$c_A = (-1)^n \left(t^n + \sum_{i=0}^{n-1} c_i t^i \...
0
votes
1answer
25 views
Calculating determinant using eigenvalues (real and complex)
I was trying to find the eigenvalues for the following characteristic polynomial:
$$p_M(\lambda)=(\lambda^{4}+1)(\lambda^2-4)$$
For this I solved the equation $p_M=0$ and I found the following two ...
0
votes
1answer
24 views
Finding characteristic polynomial and eigenvalues of a linear transformation
Let $T:R_3[x] \to R_3[x]$ Linear transformation such that:
$$
T(ax^2 + bx + c) = (a+b+c)x^2 + (2a + 2b + 2c)x + a+b-c
$$
I want to find eigenvalues for $T$.
Therefore I looked at the representing ...
1
vote
2answers
71 views
Derivative of coefficients of characteristic polynomial
Let $ X(t) $ be a square matrix of dimension $ n $. The Jacobi formula expresses the derivative of the determinant of $X(t)$ in terms of the derivative of the matrix itself. Is there an analogous ...
1
vote
0answers
32 views
Monic polynomial and companion matrix
Problem
Let $p(T) := T^n-\alpha_{n-1}T^{n-1}-\alpha_{n-2}T^{n-2}-\cdots-\alpha_0 \in K[T]$.
Additionally we have the companion matrix of $p$
$$A:= \begin{pmatrix}
0 & 1 & 0 & 0 & \...
18
votes
4answers
1k views
Can we deduce the characteristic polynomial for this matrix?
Given a square $n \times n$ matrix $A$ that satisfies $$\sum\limits_{k=0}^n a_k A^k = 0$$ for some coefficients $a_1, a_2, \dots, a_n,$ can we deduce that its characteristic polynomial is $\sum\...
2
votes
1answer
28 views
Solving for the closed form of recurrence relations using characteristic polynomial
I know how to find the closed form of some recurrence relations such as
those that are similar to the Fibonacci Sequence. I am not sure how to solve a recurrence relation using the characteristic ...
3
votes
2answers
49 views
Let $A$ be an $n \times n$ real matrix with $n \geq 2$ and characteristic polynomial $x^{n-2}(x^2-1)$, then
$\newcommand{\rank}{\operatorname{rank}}$Let $A$ be an $n \times n$ real matrix with $n \geq 2$ and characteristic polynomial $x^{n-2}(x^2-1)$, then
$A^n=A^{n-2}$
$\rank(A) \geq 2$
$\rank(A) = 2$
...
0
votes
3answers
77 views
Let $A$ be a $3\times 3$ matrix with characteristic polynomial $x^3-3x+a$, for what values of $a$ given matrix must be diagonalizable.
Let $A$ be a $3\times 3$ matrix with characteristic polynomial $x^3-3x+a$. For what values of $a$ given matrix must be diagonalizable.
I am talking about diagonalizability over reals.
Efforts:
If a ...
2
votes
1answer
150 views
How to compute the characteristic polynomial of a companion matrix to a polynomial with matrix-valued coefficients?
Consider we have a polynomial $p = z^m + b_{m-1}z^{m-1} + \dotsb + b_0$ with matrix coefficients $b_i \in M_n(\mathbb{C})$. Then we might consider the companion matrix
$$T = \left[
\begin{matrix}
0_n &...
4
votes
1answer
42 views
Question involving characteristic polynomial of a linear transformation
I was wanting some hints on a question and I have no idea how to approach this:
Suppose $F$ is a field, $V$ is an $F$-vector space and $T: V \rightarrow V$ is a linear map. Suppose $p(x) \in F[x]$ ...
0
votes
4answers
76 views
Find the characteristic polynomial |$\lambda - AI $| for this $5 \times 5$ matrix
Find the characteristic polynomial |$\lambda - AI $| for the $5\times 5$ matrix
$$A=\left(\begin{matrix}
0 & 1 & 0 & 0 & 0\\
0 & 0 & 1 & 0 & 0\\
0 & ...
4
votes
1answer
68 views
Factorizing a polynomial.
Factorize the polynomial
$$P(x)= \begin{vmatrix}
a_1^2-x & a_{1}a_2 & a_1a_3 & \cdots & a_1a_n \\
a_2a_1 & a_2^2-x & a_{2}a_3 & \cdots & a_2a_n\\
a_3a_1 & ...
1
vote
2answers
69 views
How to determine the characteristic polynomial of the $4\times4$ real matrix of ones? [duplicate]
$$\left[\begin{array}{l}1&1&1&1\\1&1&1&1\\1&1&1&1\\1&1&1&1\end{array}\right]$$
I am having difficulties with calculating this. Currently I'm stuck at ...
1
vote
2answers
170 views
How to conclude that the minimal polynomial is the characteristic?
I am given the following matrix
$$A=\begin{bmatrix}
0 & 0 & 4 & 1\\
0& 0 & 1 & 4\\
4 & 1 & 0 &0\\
1 & 4 & 0 & 0
\end{bmatrix}$$
And I have to find the ...
3
votes
2answers
95 views
Calculate the determinant of $A-2A^{-1}$ given the characteristic and minimal polynomials of $A$
Given the characteristic polynomial and minimal polynomial of $A$ being $(x+2)^{6}(x-1)^{3}$ and $(x+2)^{2}(x-1)^{3}$, respectively. How do I determine the characteristic polynomial and minimal ...
1
vote
1answer
91 views
Polynomials $\&$ Matrices
Assume $A$ is a matrix of order $n$. We know that the characteristic polynomial of matrix $A$ is obtained as follows
$$
P(x)=\det (A-x\,I)\, .
$$
Where $I$ is an identity matrix of order $n$. What ...
3
votes
3answers
728 views
Cyclic Modules, Characteristic Polynomial and Minimal Polynomial
Suppose that $\mathrm{dim}_{F}M<\infty$ for $F$ a field and $M$ an $F$ vector space. Let $T$ be a linear transformation on $M$. Show that $M$ is cyclic (as an $F[x]$ module) if and only if $m(x)$ ...
10
votes
3answers
7k views
Determining a matrix from its characteristic polynomial
Let $A\in\mathcal{M}_{n}(K)$, where $K$ is a field. Then, we can obtain the characteristic polynomial of $A$ by simply taking $p(\lambda)=\det(A-\lambda I_n)$, which give us something like
$p(\...
1
vote
2answers
1k views
Eigenvalues of a companion matrix
I've been tasked with the following:
Show that the companion matrix $C(p)$ of $p(x) = x^2 + ax + b$ has characteristic polynomial $\lambda^2 + a\lambda + b$.
Show that if $\lambda$ is an ...
0
votes
2answers
429 views
Characteristic polynomial of a square matrix of size $n$ with all entries equal to $1$
Let $A$ be a square matrix of size $n$ such that all the entries of $A$ are $1$. Find the characteristic polynomial of $A$.
0
votes
2answers
176 views
Eigenvalues of negative companion matrix
Here's a homework question I've been stuck on for a while.
Given
$$A = \begin{bmatrix}
0 & 0 & 0 & \cdots & 0 & a_0 \\
-1 & 0 & 0 & \cdots & 0 & a_1 \\
...
37
votes
2answers
1k views
Which polynomials are characteristic polynomials of a symmetric matrix?
Let $f(x)$ be a polynomial of degree $n$ with coefficients in $\mathbb{Q}$. There are well-known ways to construct a $n \times n$ matrix $A$ with entries in $\mathbb{Q}$ whose characteristic ...
3
votes
3answers
1k views
Finding the associated matrix of a linear transformation to calculate the characteristic polynomial
Let $T : M_{n \times n}(\Bbb R) \to M_{n \times n}(\Bbb R)$ be the function given by $T(A)=A^t$ (the transpose of $A$).
I need to find the minimal polynomial and the characteristic polynomial of $T$. ...
10
votes
2answers
8k views
Characteristic polynomial of an inverse
Given the characteristic polynomial $\chi_A$ of an invertible matrix $A$, I'm to find $\chi_{A^{-1}}$.
I can see that this is theoretically possible. $\chi_A$ uniquely determines the similarity class ...
0
votes
4answers
224 views
Sufficient condition for a polynomial to be a characteristic polynomial
Let $A\in \operatorname{Mat}_{n\times n}(F),~F$ being a field, satisfies $p(x)\in F[x]$ where $\deg p(x)=n$ and $p(x)$ is a monic polynomial. Can we say $p(x)=\chi_A(x)?$
41
votes
6answers
13k views
Do $ AB $ and $ BA $ have same minimal and characteristic polynomials?
Let $ A, B $ be two square matrices of order $n$. Do $ AB $ and $ BA $ have same minimal and characteristic polynomials?
I have a proof only if $ A$ or $ B $ is invertible. Is it true for all cases?
20
votes
1answer
593 views
What do characteristic polynomials characterize?
Let $R$ be an integral domain and $F$ a finitely generated free module over $R$. For a linear transformation $\alpha\in\operatorname{End}_R(F)$, the characteristic polynomial is \begin{equation}
p_\...
3
votes
1answer
938 views
When is a matrix similar to the companion matrix of its characteristic polynomial?
Let $A$ be a complex matrix and $A_c$ the companion matrix of its characteristic polynomial. From what I have read, I believe the following two statements to be true:
not every $A$ is similar to $...
32
votes
2answers
34k views
Minimal polynomials and characteristic polynomials [duplicate]
I am trying to understand the similarities and differences between the minimal polynomial and characteristic polynomial of Matrices.
When are the minimal polynomial and characteristic polynomial the ...
8
votes
5answers
902 views
Cool/Useful Examples of Characteristic and Minimal Polynomials?
I'm teaching a Linear Algebra II undergrad course and for the section on characteristic & minimal polynomials, I really don't want to just give the students a bunch of matrices that have no ...
24
votes
1answer
28k views
Elegant proofs that similar matrices have the same characteristic polynomial?
It's a simple exercise to show that two similar matrices has the same eigenvalues and eigenvectors (my favorite way is noting that they represent the same linear transformation in different bases).
...
36
votes
3answers
13k views
When are minimal and characteristic polynomials the same?
Assume that we are working over a complex space $W$ of dimension $n$. When would an operator on this space have the same characteristic and minimal polynomial?
I think the easy case is when the ...
6
votes
1answer
12k views
Characteristic polynomial of companion matrix [duplicate]
I have a matrix in companion form, $A=\begin{pmatrix} 0 & \cdots & 0& -a_{0} \\ 1 & \cdots & 0 & -a_{1}\\ \vdots &\ddots & \vdots &\vdots \\ 0 &\cdots & ...
24
votes
2answers
2k views
Do characteristic polynomials exhaust all monic polynomials?
Let $A$ be an $n\times n$ matrix, then $\mathrm{char}_A(x):=\det(xI-A)$ is a monic polynomial of degree $n$. It is called the characteristic polynomial of $A$. My question is the converse:
Let $p(x)...
23
votes
4answers
4k views
Interpreting the Cayley-Hamilton theorem
The statement of the Cayley-Hamilton Theorem is fairly straight-forward.
I now know how to find characteristic polynomials from a given matrix (or at least a matrix with certain properties that I am ...
23
votes
3answers
6k views
Coefficients of characteristic polynomial of a matrix
For a given $n \times n$-matrix $A$, and $J\subseteq\{1,...,n\}$ let us denote by $A[J]$ its principal minor formed by the columns and rows with indices from $J$.
If the characteristic polynomial of $...
