All Questions
Tagged with characteristic-polynomial linear-transformations
40 questions
0
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Prove that the minimal and the characteristic polynomial of a linear operator are the same
Suppose $V$ is an $n$ dimensional vector space over a field $F,$ and $B=\{v_1,v_2,...,v_n\}$ be an ordered basis. Let $T:V\to V$ be the linear operator such that $T(v_1)=v_2, T(v_2)=v_3,...,T(v_{n-1})=...
- 3,450
1
vote
1
answer
104
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Given Ker(T) and Ker(T-2I), is T diagonalizable?
I'm stuck on this problem that I found on my book of linear algebra.
Be $T:\mathbb{R}ˆ3 \rightarrow \mathbb{R}ˆ3$ a linear map such that
$$Ker(T-2I) = \{(x,y,z) | x+y=0\} $$ and $$Ker(T) = \{ (x,y,z)|...
- 11
0
votes
1
answer
63
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Prove dimension of kernel of irreducible quadratic factor of characteristic polynomial is less then doubled multiplicity of respective complex root
Suppose characteristic polynomial of $\varphi : V \to V$, $V$ is over $\mathbb{R}$, is written as $\chi = (t - \lambda)^k(t - \bar\lambda)^k p(x)$, where $\lambda \in \mathbb{C} \backslash \mathbb{R}$ ...
0
votes
1
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44
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Implication from definition of characteristic polynomial
I know that the characteristic function of a linear map $T:V\to V$ is defined as $\chi_T(x):=\chi_A(x)$ where $A$ is any matrix for $T$ w.r.t. some basis of $V$. I know this is well-defined as it is ...
- 477
1
vote
1
answer
283
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Minimal polynomial and characteristic polynomial of $T$
Let 𝑽 be a vector space over the field 𝑭 and 𝑻 be a linear operator
on 𝑽. Then all eigen values of 𝑻 are zeros of the minimal polynomial of 𝑻.
Minimal polynomial divides the characteristic ...
- 1,046
2
votes
1
answer
104
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Proof in infinite-dimensional space
Let $a \neq b\neq c \neq a$ be distinct real numbers, and let $f\colon E \to E$ be an endomorphism
of a real vector space $E$ such that
$$
(f − aI)(f − bI)(f − cI) = 0.
$$
Show that
$$
E = \ker(f − aI)...
- 477
2
votes
1
answer
69
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Prove or disprove if $m(x)=\prod_{j=1}^{s}(x-\lambda_{j}^{2})$ is the minimal polynomial of a linear map $T\circ T:V\to V$
Prove or disprove:
If $m(x)=\prod_{j=1}^{s}(x-\lambda_{j}^{2})$ (with $\lambda_i\not=\lambda_j$ for $i\not=j$) is the minimal polynomial of a linear transformation $T\circ T:V\to V$, with $V$ a $\...
- 418
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5
answers
166
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Suppose that $f: V \to V$ is a $k$-linear transformation such that $f^m = 0$ for some integer $m.$ Prove that $f^n = 0.$
Here is the question I want to tackle:
Let $k$ be a field and let $V$ be an $n$-dimensional vector space over $k.$ Suppose that $f: V \to V$ is a $k$-linear transformation such that $f^m = 0$ for some ...
user965463
1
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0
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127
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"Linear algebraic" proof of Frobenius normal form
Theorem: Let $\mathcal{A}$ be a linear operator on a finite dimensional vector space $V$, there exists a basis such that $V$ can be represented by the direct sum of some $\mathcal{A}$-cyclic subspace ...
- 59
4
votes
1
answer
453
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Characteristic polynomial of projection
Let $V$ be a $n$ dimensional vector space over a field $\mathbb{k}$, and let $P:V \rightarrow V$ be a linear map such that $P^2=P$. i.e., A linear map is a projection.
I want to find all possible ...
- 6,540
1
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0
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135
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Other invariants of a linear transformation?
The minimal and characteristic polynomials of a linear transformation encode much useful information about the transformation. Of course, they do not encode all useful information -- for example, we ...
- 3,392
1
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1
answer
387
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generic matrix similar to companion matrix of its minimal polynomial
Write $A$ for the generic matrix (comprised of indeterminates). In their constructive commutative algebra book, Lombardi and Quitte write that since the determinant of the family $(e_1,Ae_1,\dots,A^{n-...
- 14.2k
1
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1
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125
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Conjugacy of linear transformations with the same characteristic polynomial
Consider the following result and proof about conjugacy of linear transformations:
Here $E$ and $F$ are $n$-dimensional vector spaces, $\varphi:E\to E$ and $\psi:F\to F$. The $\mu$'s are the minimal ...
- 3,392
1
vote
2
answers
127
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Prove that $P^k=P$ for any $k \in \mathbb{N}$ where $1$ is the only eigenvalue of $P$ implies $P=I$.
I'm having trouble proving this, using the fact $P^k=P$.
($P \in L(V)$ where $V$ is a finite-dimensional complex vector space.)
Here's my work (I didn't use $P^k=P$, but it still looks valid to me ...
- 1,298
0
votes
1
answer
122
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Linear Transformation $T:\mathbb{R}^3 \rightarrow \mathbb{R}^3$, $T^3=T$, $T^2 \neq T$, $T^2\neq I$
I need to show that the linear transformation $T:\mathbb{R}^3 \rightarrow \mathbb{R}^3$, $T^3=T$, $T^2 \neq T$, $T^2\neq I$ is such that
(1) $\text{null}\,T=1$ or $2$
(2) $\text{null}\,T=2$ $\...
- 724
2
votes
3
answers
73
views
Prove that $A$ in invertible through characteristic polynomial: $x^{500}+x^{100}-x+4$
Let $A$ be a matrix with charecteristic polynomial $$p(x)=x^{500}+x^{100}-x+4$$
Prove that $A$ is invertible.
I'm very lost with this one, because I don't know how to calculate the eigenvalues, I ...
- 577
0
votes
1
answer
48
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Linear transformations of a specific characteristic polynomial. [closed]
I want to know all linear transformations having the characteristic polynomial $(x-1)^3(x+1)^2$?how can I know their number? how can I know them exactly? Is it by Jordan blocks or what?
Any help will ...
user889696
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1
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49
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If we're given characteristic and minimal polynomial of a linear transformation, how can we find all its possible Jordan forms?
Determine all possible Jordan forms of a linear transformation with characteristic polynomial $(x−2)^4(x−3)^3$ and minimal polynomial $(x−2)^2(x−3)^2$.
1
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1
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192
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Minimal and characteristic polynomials of orthogonal transformations
I am given orthogonal linear transformation $U: \mathbb{R}^4 \to \mathbb{R}^4$, represented by
$$A=\begin{bmatrix} 1/2 & 1/2 & 1/2 & -1/2 \\ 1/2 & 1/2 & -1/2 & 1/2 \\ 1/2 & ...
- 21
3
votes
0
answers
119
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Characteristic polynomials of two matrices containing the same elements
I asked this question in a slightly different form and didn't receive any comments. Two real, symmetric, positive semidefinite matrices $A$ and $B$ contain the same elements (in different orders). ...
- 31
2
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0
answers
201
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Given a 4x4 Singular Matrix. find characteristic polynomial
I'm trying to solve this issue:
Given A, a 4x4 singular Matrix. It is known that $\rho(A+2I)=2$ and $|A-2I| =0$.
Find the characteristic polynomial of A, is A similar to a diagonal matrix?
I've ...
- 411
1
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1
answer
150
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Computing the characteristic polynomial of a linear transformation
I am confused as to how to solve this problem, as the linear transformation includes an arbitrary inner product.
Let $V$ be a finite-dimensional inner product space. Fix vectors $v,w \in V$ and ...
user862302
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1
answer
66
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Relation between characteristic polynomials of $A$ and the linear map $\mathscr{L}(X)=AX$
For a fixed $A \in M_n (\mathbb{R})$, define $\mathscr{L}(X)=AX$ on the set $M_n (\mathbb{R})$. We have to find how characteristic polynomials of $\mathscr{L}$ and $A$ are related. I need to get my ...
- 855
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450
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Irreducible Polynomial as Characteristic polynomial of a Matrix.
Given an irreducible polynomial $p$ of degree $n$ in $\mathbb{R}[X]$, does there exists an $n\times n $ real matrix with $p$ as its characteristics polynomial.
Since similar matrices have the same ...
- 1,657
1
vote
2
answers
122
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Prove that If $f$ and $h$ commute and $h$ is nilpotent, then $f+h$ and $f$ have the same characteristic polynomial
let $V$ be a finite dimensional vector space over $\mathbb{K}$ ($\mathbb{R}$ or $\mathbb{C}$).
and $f$ and $h$ two vector space endomorphisms of $V$, such as that : $h$ is nilpotent and $f \circ h = h\...
- 523
1
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1
answer
40
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The minimal poly of $\sqrt[3]{2}$ over $\Bbb{Q}$ is equal to $\det(T_a - xI)$ where $T_a$ is a matrix over $\Bbb{Q}$ that represents mult. by $a$.
Let $K/F$ be a field extension of degree $n \in \Bbb{N}$ and for each $a \in K$ define $L_a(x) = a x$. Then $L_a(x)$ is an $F$-linear transformation of $K$ as a vector space of dimension $n$. So ...
- 21.6k
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0
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157
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Do factors in operator polynomial commute?
The problem I am working on asks: Let $A$ be a complex $n \times n$ matrix and suppose $y$ is a nonzero complex $1 \times n$ vector. Show that there is a nonzero polynomial $q(x)$ and an eigenvalue $\...
- 2,499
3
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1
answer
149
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The minimal and characteristic polynomial of the linear operator $T_P(M)=PMP^{-1}$ acting on $S(3)$
For an invertible matrix $P\in\mathbb{R}^{n\times n}$ let
$T_P\colon\mathbb{R}^{n\times n}\to\mathbb{R}^{n\times n}$ be the
linear map defined by $T_P(M)=PMP^{-1}$ for any
$M\in\mathbb{R}^{n\times n}$....
- 2,345
0
votes
1
answer
91
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Show that two triagonalizable matrices $A$ and $B$ are similar iff $\chi_A=\chi_B$ and $\mu_A=\mu_B$
Let $K$ be a field, $n\ge 1\in\mathbb{N}_{\ge 1}$ and let $A,B\in M_n(K)$ be two triagonalizable matrices, such that for every eigenvalue $\lambda$ of $A$ there holds that $\dim V(\lambda,A)^{\text{...
user731634
1
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1
answer
202
views
Absolute value of coefficients of the characteristic polynomial of a unitary matrix
Let $A$ be a unitary $n\times n$ matrix. Show that $|c_k|\leq\binom{n}{k}$ for the coefficients of the characteristic polynomial of A, $\chi_A(X)=c_0+c_1X+\ldots +c_{n-1}X^{n-1}+X^n$.
I know that $\...
user731634
0
votes
1
answer
94
views
Characteristic polynomial of unitary operator [closed]
If I have $f_T(x)=x^2-1$ is that mean that my $T$ is unitary? I tried to show that is not true but I didn't succed so I might missing smoething
- 1,430
0
votes
1
answer
60
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Find characteristic polynomial of $T^{-1}$
Given that $[T]_B=\begin{bmatrix}1&0\\0&2\end{bmatrix}$ for some linear transformation $T:V\rightarrow V$, and $B$ basis for $V$, I'm trying to find the characteristic polynomial of $T^{-1}$.
...
- 559
1
vote
2
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733
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How to show that $det(AB-xI)=det(BA-xI)$ ,for any $x\in \mathbb F$.
There is a problem in Hoffmann Kunze:
Show that $AB$ and $BA$ have the same characteristic polynomials,where $A,B$ are both $n\times n$ matrices in $\mathbb F$.
If $A$ or $B$ is invertible it can be ...
- 3,801
1
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1
answer
834
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Proving characteristic polynomial and invertibility
Let A be an n by n matrix with characteristic polynomial $f(t)=(-1)^n t^n+a_{n-1}t^{n-1}+...+a_1 t+a_0$. Prove that $f(0)=a_0=det(A)$. deduce that A is invertible if and only if $a_0 \neq 0$.
To ...
- 543
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votes
4
answers
35
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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}$ ...
user731634
1
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1
answer
272
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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)$ with ...
user615817
0
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1
answer
52
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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 ...
- 1,657
4
votes
1
answer
55
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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 $p(x) \in F[x]$ ...
- 343
3
votes
3
answers
2k
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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 $T$. ...
- 2,456
4
votes
2
answers
2k
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Finding an operator that satisfies a given Minimal and characteristic Polynomial?
I am studying for an upcoming Linear Algebra exam. I am going through the questions from an old exam the instructor gave out, and I have come to this problem:
Give an example of an operator on a ...
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