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.

Filter by
Sorted by
Tagged with
0 votes
0 answers
13 views

$A ∈ GL_n(K)$. Show that there exists a polynomial $P(X) ∈ K[X]$ of degree $< n$ such that $A^{−1} = P(A)$ [duplicate]

Hey I am having problems with this exercise. Can someone help me? Let $A ∈ GL_n(K)$. Show that there exists a polynomial $P(X) ∈ K[X]$ of degree $< n$ such that $A^{−1} = P(A)$. To show that there ...
0 votes
0 answers
60 views

Why is the following "proof" false? [duplicate]

Exercise: Why is the following "proof" false? $\text{ch}_A(xI-A)=\det(xI-A)$ Substitude $A$ for $x$, and obtain $\det(A-A)=0=\text{ch}_A(A)$. Solution: Explanation We cannot substitude $A$ ...
0 votes
0 answers
32 views

Proof that the characteristic polynomial of a matrix can be expressed as a sum of the matrix's determinant and scaled powers of its eigen values

In the book, Introduction to Computational Linear Algebra, the authors state the following: For any matrix $A \in \mathbb{R}^n$, its characteristic polynomial, $p_\text{A}(\lambda) = \text{det}(A - \...
0 votes
0 answers
28 views

Characteristic polynomials of induced linear maps

Take an $n \times n$ matrix $A$ with its characteristic polynomial $p(t) = Det(A-tI)$. There are a variety of induced linear functionals $f : M_{n,n}(\mathbb R) \to M_{n,n}(\mathbb R)$ such as $H \...
  • 22.6k
3 votes
3 answers
61 views

How to find ${\rm rank}(2I_n-A)$ where A is a square matrix of size $n$ and $A^3 - 6A^2 + 12A = 0_n$?

I think the rank has to be $n$ since anything else would be impossible to prove with so little information about the matrix. $$\det(2I_n-A) = -P_A(2) = -\det(A - 2I_n) \ .$$ So, if I can show the ...
2 votes
1 answer
72 views

What are the eigenvalues of this arrowhead matrix?

Suppose $p_0,\ p_1,\ \dots,\ p_q$ are positive such that $p_0+p_1+\dots+p_q=1$. I am wondering how to find the eigenvalues of the following arrowhead matrix $$A=\begin{bmatrix} 1 & p_1 & \dots ...
0 votes
1 answer
36 views

Prove dimension of kernel of irreducible quadratic factor of characteristic polynomial is less then doubled multiplicity of respective complex root

Suppose characteristic polynomial of $\varphi : V \to V$, $V$ is over $\mathbb{R}$, is written as $\chi = (t - \lambda)^k(t - \bar\lambda)^k p(x)$, where $\lambda \in \mathbb{C} \backslash \mathbb{R}$ ...
1 vote
1 answer
67 views

The rank of the characteristic polynomial of a $\lambda$-matrix

Assume $F(\lambda, y) = \mathrm{det}(yI - V)$, where $V = (V_{ij}(\lambda))_{4\times 4}$ and each $V_{ij}(\lambda)$ is a polynomial of $\lambda$. If $F(\lambda, y) = 0$ always has four different roots ...
0 votes
0 answers
112 views

Show that $rank(A)=rank(A^{k})$ for every $k\geq1$ if and only if there exists $m\geq2$ such that $A^{m}=A$

We have $M\left(X\right)=\left\{X^{k}|k\in \mathbb{N}^{*}\right\}$ for every $(n×n)$ matrix $X$ with complex entries. If $A$ is $(n×n)$ matrix with complex entries and $M\left(A\right)$ is finite, ...
0 votes
1 answer
79 views

Show that there exists $A$ $(n×n)$ matrix with complex entries such that...

We have $M\left(X\right)=\left\{X^{k}|k\in \mathbb{N}^{*}\right\}$ Show that there exists $A$ $(n×n)$ matrix with complex entries such that $M\left(A\right)$ is finite and it does not contain $0$ and $...
0 votes
1 answer
30 views

Characteristic polynomial of real matrix with complex eigenvalues

I need some clarification regarding the characteristic polynomial for a real matrix with complex eigenvalues. I am given the matrix $$A=\begin{pmatrix} -1 & -5 & 4 \\ 1 & 1 & -1 \\ 0 &...
1 vote
0 answers
20 views

Best (or correct) approach on finding characteristic polynomial of this $4 \times 4$ matrix

Let $f: \mathbb{R}^4 \to \mathbb{R}^4$ be a linear transformation with matrix representation \begin{align*} A = \begin{pmatrix} -5 & -5 & -9 & 7 \\ 8 & 9 & 18 &...
  • 2,646
1 vote
1 answer
96 views

Show that $A+B=AB+BA$ iff $\text{Tr}(A)=\text{Tr}(B)=\text{Tr}(AB)=1$

We have $A,B$ $(2×2)$ matrices with complex entries. We know $AB≠BA$. Show that $A+B=AB+BA$ if and only if $\text{Tr}(A)=\text{Tr}(B)=\text{Tr}(AB)=1$. I tried writing $A=X+Y$ and $B=X-Y$ so we can ...
4 votes
2 answers
276 views

Show that $\det(xA+yB+zI_{n})=\det(yA+xB+zI_{n})$

We have $A$ and $B$ $(n×n)$ matrices with complex entries. We know that $A-B=AB-BA$. Show that $$\det(xA+yB+zI_{n})=\det(yA+xB+zI_{n})$$ for every $x,y,z$ complex numbers with $x+y≠0$. We can see that ...
0 votes
0 answers
69 views

Prove that $P(A)=a_nA^n+a_{n-1}A^{n-1}+...+a_1A+a_0I=0$ if $A$ is diagonalizable.

The question goes as: Let A be a diagonalizable square matrix with N rows and N columns. Let $p(x)=a_nx^n+a_{n-1}x^{n-1}+...+a_1x+a_0$ be the characteristic polynomial of A. Prove: $p(A)=a_nA^n+a_{n-...
1 vote
3 answers
90 views

Show that matrix $A+I_{3}$ is invertible if $A$ is orthogonal with $\operatorname{trace}(A) > 1$

We have $A$ $(3×3)$ matrix with real entries. We know that A is orthogonal and $\operatorname{trace}(A)>1$. Show that matrix $A+I_{3}$ is invertible. We can see that $\det(A)=1$ or $\det(A)=-1$. We ...
0 votes
0 answers
23 views

Find the characteristic polynomial of $B \mapsto AB + BA$ [duplicate]

Let $V$ be the set of 2x2 complex matrices. Let $T_A : V \rightarrow V$ send $B \mapsto AB + BA$. We want to find the characteristic polynomial of $T_A$, $\chi_{T_A}(t)$, in terms of $\det{A}$ and $\...
-2 votes
1 answer
44 views

Show that the sum of the elements on every line of matrix $A^{-1}$ is $s^{-1}$. [closed]

We have $A$ $(n×n)$ matrix with complex entries. We know that $\det(A)≠0$ and the fact that the sum of the elements on every line is s. Show that the sum of the elements on every line of matrix $A^{-1}...
3 votes
2 answers
133 views

Show that $\det(A)=2^{p}$

We have a $(n×n)$-matrix $A$ with complex entries such that $\,A^{2}=3A-2I$.$~$ Show that there exists $p\in\{0,1,2,...,n\}$ such that $\det(A)=2^{p}$. I don't know if my proof is good. I took the ...
1 vote
1 answer
36 views

Jordan Normal form- order of eigenvalues in the diagonal

The following matrix is given: After doing all the usual calculations I got the following polynomial and eigenvalues Now we know that for the eigenvalue of 1, the algebraic multiplicity is going to ...
1 vote
0 answers
20 views

Cayley Hamilton theorem and divisibility in the reverse direction

Let $A$ be an $n \times n$ matrix with entries in some field $K$. Call $\chi_A$ its characteristic polynomial and $\mu_A$ its minimal polynomial. Cayley Hamilton theorem states that $\mu_A$ divides $\...
2 votes
1 answer
52 views

Coefficients of characteristic polynomial and eigenvalues

Let $A$ be a $3x3$ symmetric (thus, orthogonally diagonalizable) real matrix. We know that its characteristic polynomial is (in the variable $λ$) $$-λ^3 + tr(A)λ^2-cλ+det(A)$$ where $c\in \mathbb{R}$ ...
  • 102
1 vote
2 answers
142 views

Show that equation $\det(A+xB)=0$ has real solutions if and only if $\det(A^{2}+B^{2})\geq(\det(A)+\det(B))^{2}$

We have $A,B$ two $2×2$ matrices with real values and we know $\det(AB-BA)=0$. Show that equation $\det(A+xB)=0$ has real solutions if and only if $$\det(A^{2}+B^{2})\geq(\det(A)+\det(B))^{2}.$$ I ...
7 votes
1 answer
201 views

Prove $\det((AB)^{n}-(BA)^{n})$ is a perfect cube.

We have $A,B$ two $3×3$ matrices with integer numbers. We know that $(AB)^{2}+BA=(BA)^2+AB$. a) Show that $\det((AB)^{n}-(BA)^{n})$ is divisible by $det(AB-BA)$. b) Show that if $\det(AB-BA)=1$, then $...
3 votes
0 answers
84 views

Theorem 4 (Cayley-Hamilton), Section 6.3 of Hoffman’s Linear Algebra

Let $T$ be a linear operator on a finite dimensional vector space $V$. If $f$ is the characteristic polynomial for $T$, then $f(T)=0$; in other words, the minimal polynomial divides the characteristic ...
  • 3,188
0 votes
1 answer
39 views

Implication from definition of characteristic polynomial

I know that the characteristic function of a linear map $T:V\to V$ is defined as $\chi_T(x):=\chi_A(x)$ where $A$ is any matrix for $T$ w.r.t. some basis of $V$. I know this is well-defined as it is ...
  • 303
3 votes
2 answers
65 views

Example 4, Section 6.3 of Hoffman’s Linear Algebra

Example 4: In Example $2$, the operator $T$ also had the characteristic polynomial $f=(x-1)(x-2)^2$. But, this $T$ is not diagonalizable, so we don’t know that the minimal polynomial is $(x-1)(x-2)$. ...
  • 3,188
0 votes
2 answers
54 views

Linear algebra — similarity of reflection matrices

The question is as follows: Let $A,B$ be $2\times2$ reflection matrices. Are $A$ and $B$ similar? What I’ve tried: It did seem like a proof to me: I have calculated the characteristic polynomial and ...
0 votes
1 answer
72 views

Exercise 9, Section 6.3 of Hoffman’s Linear Algebra

Let $A$ be an $n \times n$ matrix with characteristic polynomial $$f=(x-c_1)^{d_1}\cdots (x - c_k)^{d_k}$$ Show that $$c_ld_1+\cdots + c_kd_k=\text{trace} (A)$$ My attempt: Characteristic polynomial ...
  • 3,188
0 votes
0 answers
68 views

Exercise 8, Section 6.3 of Hoffman’s Linear Algebra

Let $P$ be the operator on $R^2$ which projects each vector onto the $x$-axis, parallel to the $y$-axis: $P(x,y)=(x, 0)$. Show that $P$ is linear. What is the minimal polynomial for $P$? My attempt: ...
  • 3,188
0 votes
2 answers
65 views

Exercise 2, Section 6.3 of Hoffman’s Linear Algebra [duplicate]

Let $a$, $b$, and $c$ be elements of a field $F$, and let $A$ be the following $3\times 3$ matrix over $F$: $$A=\begin{bmatrix}0 & 0& c\\ 1& 0& b\\ 0& 1& a\\ \end{bmatrix}$$ ...
  • 3,188
7 votes
3 answers
288 views

Show that $\det(AB-BA) × \det(AC-CA) \geq 0$ if $A^2 = -BC$

We have $A,B,C$ three $n×n$ matrices with real entries. We know that $$ A^2 = -BC $$ and we want to show that $$ \det(AB-BA) × \det(AC-CA) \geq 0 \,. $$ We can easily show that for $n=2k$ we have $...
0 votes
1 answer
44 views

Range Space and Null Space of Projection Matrix

I'm studying my midterm exam and solving the problem set. Unfortunately, there is no solution manual for this set. I will show what I did and I will ask my specific question. Firstly, the question is ...
0 votes
0 answers
41 views

How is the matrix $(sI - A)$ called whose determinant is the characteristic polynomial of a (real valued) matrix $A$?

Context: When calculating eigenvalues of a real valued matrix $A$ one often constructs an auxiliary polynomial matrix $M(s):= (sI - A)$ and then calculates its determinant $d(s):=\det\Big(M(s)\Big)$ – ...
  • 407
1 vote
1 answer
43 views

Minimal polynomial and characteristic polynomial of $T$

Let 𝑽 be a vector space over the field 𝑭 and 𝑻 be a linear operator on 𝑽. Then all eigen values of 𝑻 are zeros of the minimal polynomial of 𝑻. Minimal polynomial divides the characteristic ...
0 votes
1 answer
26 views

Show that characteristic polynomial of $Ty = ay$ is power of minimal polynomial of $a$

Let $K/F$ be a finite, separable, algebraic field extension and let $T: K\to K, Ty = ay$. Show that $p = m^n$ where $p$ is $T$'s characteristic polynomial and $m$ is $a$'s minimal polynomial. $m$ ...
  • 13
-1 votes
1 answer
51 views

Prove that A^k = 0 implies A^2 = 0 for A_2x2 [duplicate]

I shall prove the statement that if A is a 2x2 matrix and $A^2 \neq 0$, then it follows that $A^k \neq 0$ for $k > 2$ as well. I intended to prove the contraposition, but I’m not quite sure how to ...
2 votes
0 answers
29 views

Why does the character of the alternating $2$-tensor representation appear in characteristic polynomial of a $3 \times 3$ matrix?

Fact one: If $V$ is a three-dimensional vector space, consider the matrix representation $A$ of a linear transformation from $V$ to $V$. Then the characteristic polynomial of the $3 \times 3$ matrix $...
0 votes
0 answers
30 views

Relationship between generalized characteristic and trace coefficients?

Let $E$ be an $n$-dimensional vector space over a field $\Gamma$ of characteristic zero. If $\varphi$ is a linear transformation of $E$, define $$C_p(\varphi)=\mathop{\mathrm{tr}}(\textstyle\bigwedge^...
  • 2,245
1 vote
1 answer
164 views

If $A$ commutes with $(AB - BA)^2$, is $\det(AB - BA) = 0$?

We have $A$ and $B$ are $3 \times 3$ matrices with complex numbers. We know matrix $A$ is commuting with matrix $(AB-BA)^2$. Can you show $\det(AB-BA)=0$? I tried using some Hamilton Cayley Theorem on ...
0 votes
0 answers
43 views

Exercise 6, Section 6.2 of Hoffman’s Linear Algebra

Let $T$ be the linear operator on $\Bbb{R}^4$ which is represented in the standard ordered basis by the matrix $$\begin{bmatrix} 0 & 0 & 0 & 0\\ a & 0 & 0 & 0\\ 0 & b & ...
  • 3,188
0 votes
0 answers
33 views

Characteristic polynomial of 4 by 4 function matrix

I want to find out the eigenvalues of the following matrix \begin{equation} \begin{pmatrix} e^{2k+J(p_x+p_y)} & e^{Jp_x} & e^{Jp_y} & e^{-2k-J(p_x+p_y)}\\ e^{Jp_x} & e^{2k+J(...
0 votes
1 answer
33 views

How do you get the characteristic polynomial of a recursion?

For example, the characteristic polynomial of the Fibonacci sequence is $x^2 -x -1$. What are the steps involved in this?
  • 1
1 vote
1 answer
91 views

Eigenvalues of symmetric tridiagonal matrices with complex entries

In this paper the authors proved that for a real symmetric tridiagonal matrix $T_n$, where $b_i \neq 0$, as follows $$T_n = \begin{bmatrix} a_1&b_1&0&0&0&0&0&0&\cdots&...
  • 931
2 votes
0 answers
75 views

General solution for the characteristic polynomial of a matrix of $n\times n$ matrix sub-blocks

Let's work on the finite field $\mathbb{Z}_2$ and define the matrix S as \begin{equation} S = \begin{bmatrix} 1 & 0 & 0 & \ldots & 0 & 0 & 1 \\ 1 & 1 & 0 & ...
0 votes
1 answer
42 views

How to find the eigenvalues and their geometric multiplicities without being given a matrix

For a real $3\times3$ matrix $A$ with determinant 12 and $c_A(x) = (x − 2)m_A(x),$ how do I find the eigenvalues and their geometric multiplicities and indices? I am lost without a matrix. $\det(A)=12$...
  • 49
3 votes
2 answers
89 views

If two matrices have same characteristic polynomial, then if square root for one exists, it also exists for the other one.

This is True/False question from the recent exam. Statement: Suppose $A$ and $B$ are two elements of $M_n(\mathbb{R})$ such that their characteristic polynomials are equal. If $A=C^2$ for some $C\in ...
  • 2,447
2 votes
1 answer
60 views

Proof in infinite-dimensional space

Let $a \neq b\neq c \neq a$ be distinct real numbers, and let $f\colon E \to E$ be an endomorphism of a real vector space $E$ such that $$ (f − aI)(f − bI)(f − cI) = 0. $$ Show that $$ E = \ker(f − aI)...
0 votes
0 answers
12 views

Find representatives for all the conjugacy classes of elements of order dividing 8

Problem: Find representatives for all the conjugacy classes of elements of order dividing 8 in $\text{GL}_4(\mathbf{F}_2)$ and give the orders of the representatives. My attempt: Firstly, $\vert \text{...
-1 votes
1 answer
72 views

Showing $\lambda I_V$ diagonalizable and has only one eigenvalue

Problem: Let $V$ be a finite-dimensional vector space, and let $\lambda$ be any scalar. For any ordered basis $\beta$ for $V$, prove that $[\lambda I_V]_{\beta}=\lambda I$. Then compute the ...

1
2 3 4 5
9