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.
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$A ∈ GL_n(K)$. 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 ...
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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 $A$ ...
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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 - \...
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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 \...
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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 ...
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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
$$A=\begin{bmatrix}
1 & p_1 & \dots ...
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Prove dimension of kernel of irreducible quadratic factor of characteristic polynomial is less then doubled multiplicity of respective complex root
Suppose characteristic polynomial of $\varphi : V \to V$, $V$ is over $\mathbb{R}$, is written as $\chi = (t - \lambda)^k(t - \bar\lambda)^k p(x)$, where $\lambda \in \mathbb{C} \backslash \mathbb{R}$ ...
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The rank of the characteristic polynomial of a $\lambda$-matrix
Assume $F(\lambda, y) = \mathrm{det}(yI - V)$, where $V = (V_{ij}(\lambda))_{4\times 4}$ and each $V_{ij}(\lambda)$ is a polynomial of $\lambda$.
If $F(\lambda, y) = 0$ always has four different roots ...
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Show that $rank(A)=rank(A^{k})$ for every $k\geq1$ if and only if there exists $m\geq2$ such that $A^{m}=A$
We have $M\left(X\right)=\left\{X^{k}|k\in \mathbb{N}^{*}\right\}$ for every $(n×n)$ matrix $X$ with complex entries. If $A$ is $(n×n)$ matrix with complex entries and $M\left(A\right)$ is finite, ...
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Show that there exists $A$ $(n×n)$ matrix with complex entries such that...
We have $M\left(X\right)=\left\{X^{k}|k\in \mathbb{N}^{*}\right\}$
Show that there exists $A$ $(n×n)$ matrix with complex entries such that $M\left(A\right)$ is finite and it does not contain $0$ and $...
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Characteristic polynomial of real matrix with complex eigenvalues
I need some clarification regarding the characteristic polynomial for a real matrix with complex eigenvalues. I am given the matrix
$$A=\begin{pmatrix}
-1 & -5 & 4 \\
1 & 1 & -1 \\
0 &...
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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 &...
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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 ...
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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 ...
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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-...
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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 ...
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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 $\...
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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}...
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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 ...
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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 ...
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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 $\...
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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 $c\in \mathbb{R}$ ...
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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 ...
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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 $...
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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 ...
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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 ...
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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 $(x-1)(x-2)$. ...
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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 ...
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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 ...
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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: ...
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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 $F$: $$A=\begin{bmatrix}0 & 0& c\\ 1& 0& b\\ 0& 1& a\\ \end{bmatrix}$$ ...
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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 $...
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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 ...
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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 $d(s):=\det\Big(M(s)\Big)$ – ...
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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 ...
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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.
$m$ ...
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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 ...
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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 $...
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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
$$C_p(\varphi)=\mathop{\mathrm{tr}}(\textstyle\bigwedge^...
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If $A$ commutes with $(AB - BA)^2$, is $\det(AB - BA) = 0$?
We have $A$ and $B$ are $3 \times 3$ matrices with complex numbers. We know matrix $A$ is commuting with matrix $(AB-BA)^2$. Can you show $\det(AB-BA)=0$?
I tried using some Hamilton Cayley Theorem on ...
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Exercise 6, Section 6.2 of Hoffman’s Linear Algebra
Let $T$ be the linear operator on $\Bbb{R}^4$ which is represented in the standard ordered basis by the matrix $$\begin{bmatrix} 0 & 0 & 0 & 0\\ a & 0 & 0 & 0\\ 0 & b & ...
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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(...
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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?
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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
$$T_n = \begin{bmatrix}
a_1&b_1&0&0&0&0&0&0&\cdots&...
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General solution for the characteristic polynomial of a matrix of $n\times n$ matrix sub-blocks
Let's work on the finite field $\mathbb{Z}_2$ and define the matrix S as
\begin{equation}
S = \begin{bmatrix}
1 & 0 & 0 & \ldots & 0 & 0 & 1 \\
1 & 1 & 0 & ...
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How to find the eigenvalues and their geometric multiplicities without being given a matrix
For a real $3\times3$ matrix $A$ with determinant 12 and $c_A(x) = (x − 2)m_A(x),$ how do I find the eigenvalues and their geometric multiplicities and indices?
I am lost without a matrix. $\det(A)=12$...
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If two matrices have same characteristic polynomial, then if square root for one exists, it also exists for the other one.
This is True/False question from the recent exam.
Statement: Suppose $A$ and $B$ are two elements of $M_n(\mathbb{R})$ such that their characteristic polynomials are equal. If $A=C^2$ for some $C\in ...
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Proof in infinite-dimensional space
Let $a \neq b\neq c \neq a$ be distinct real numbers, and let $f\colon E \to E$ be an endomorphism
of a real vector space $E$ such that
$$
(f − aI)(f − bI)(f − cI) = 0.
$$
Show that
$$
E = \ker(f − aI)...
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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{...
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1
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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 ...
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