Questions tagged [jordan-normal-form]

The Jordan normal form of a matrix is a similar block matrix having diagonal blocks when the matrix is diagonalizable and diagonal + nilpotent blocks more generally.

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Describing invariant subspaces from characteristic polynomial and minimal polynomial

I am working on the following Linear Algebra problem: (a) Suppose $T: \mathbb{R}^4 \longrightarrow \mathbb{R}^4$ is a linear transformation with characteristic polynomial $x^2(x-1)$. Describe the 3-...
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Find the Jordan basis of the given matrix

$A = \begin{bmatrix} 3 & -1& -3 & 1& 0\\ 5 & -2 & -4 & -1 & 0 \\ 2 & - 1 & -2 & 1 & 0 \\ -1 & 0 & 1 & 1 & 0\\ -2 &...
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Size of Jordan block

Imagine that I'm writing the Jordan form of a matrix and I know that the eigenvalue needs to appear 4 times in the diagonal (algebraic multiplicity is 4) and we need 2 Jordan blocks (geometric ...
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Jordan Canonical Form of linear operator on vector space of polynomials

I'm working on the following problem: Let $V_{100}$ denote the $\mathbb{C}$-vector space consisting of all polynomials in $\mathbb{C}[x]$ of degree $\leq 100$. Let $T$ be the linear operator on $V_{...
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Basics of Jordan matrix, please clarify the following

Let $$J=\oplus_{i=1}^{k} J_{n_i}(\lambda)$$ where $J_{n_i}(\lambda)$ is a jordan block of size $n_i$ with $\lambda $ on its diagonal, and $\sum_{i=1}^{k}n_i = n $, so $J$ is $n\times n$ matrix, ...
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Every matrix is similar to a Jordan matrix, help finding similarity matrix

Let $A\in M_n(C)$. I want to show that it is similar to a Jordan matrix By Schur's generalization, if $A\in M_n(C)$ with $\lambda_1,\ldots\lambda_k$ distinct eigenvalues, each of multiplicity $m_i$, ...
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Basis of generalized eigenspace as a disjoint union of cycles

I am studying the Jordan canonical form from the book Linear Algebra by Friedberg, Insel, and Spence (4th Edition). I have a question regarding the following theorem: Theorem 7.7. Let $T$ be a ...
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Jordan canonical form for a 4 by 4 matrix

So, I am given the matrix $$A =\begin{pmatrix} 0 & 1 & 0 & 0 & \\ -1 & 2 & 0 & 0 & \\ -2 & 2 & 1 & 0 & \\ 0 & 1 & 0 & -1 \end{...
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Find the Jordan matrix of the operator

We know that characteristic polynomial is $P(x) = x^n$ and $$\dim \ (\ker\ a \cap \mathrm{Im}\ a) = 1$$ with $a: V\rightarrow V$. I'm not sure if we can definitely say how this matrix looks like. ...
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What are the possible Jordan canonical forms for $A$?

Let A be an (8 × 8) matrix over the complex numbers, and suppose that the characteristic polynomial is = $(2 − x)^8$ and the minimum polynomial is $(x − 2)^4$. What are the possible Jordan canonical ...
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Jordan form exercise

What am I doing wrong? I've been learning how to put matrices into Jordan canonical form and it was going fine until I encountered this $4 \times 4$ matrix: $A=\begin{bmatrix} 2 & 2 & 0 &...
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Jordan form and basis for 5 x 5 matrix

Is anybody able to explain the solution to part c of this question above? I don't understand how the characteristic polynomial was determined, or how the basis was found.
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Jordan form of a specific 2x2 matrix

I'm trying to follow an algorithm for finding the Jordan form for the matrix: $$ \begin{pmatrix} 0 & -1 \\ 4 & 4 \\ \end{pmatrix} $$ Its eigenvalue is 2 with ...
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eigenvector expansion with non symmetric matrices

I am struggling with the following question: When is it true that I can use the (right, eventually) eigenvectors of a finite dimensional matrix as a basis and thus write down an eigenvector ...
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Find the invariant factors and elementary divisors from the relations matrix.

Let $V$ be a finite dimensional vector space over $\Bbb C$, and $T$ be a linear operator on $V$. Consider $V$ as an $\Bbb C[x]$-module by defining $xv = T(v)$ for each $v \in V$. Let $$A = \begin{...
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How many similarity classes are zeroes of a given polynomial.

I am looking for an easy way of calculating the number of similarity classes of complex matrices that satisfy some polynomial $p(t)$. As an example consider $p(t)=(t-1)^3(t+1)^4$ and $5\times 5$ ...
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Possible Jordan canonical forms of identity matrix plus a nilpotent matrix

I am working on the following Linear algebra problem: Suppose that $N$ is a nilpotent $5 \times 5$ real matrix (so $N^5$ is the zero matrix). List all possible Jordan canonical forms of $I + N$. ...
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Reducing the basis vectors of $Ker(A+I)^2$ using the basis vector of $Ker(A+I)$

Please consider taking a look on the example 2 in this pdf. In the attached pdf, in example 2, the author says Reducing the basis vectors of $Ker(A+I)^2$ using the basis vector of $Ker(A+I)$, we end ...
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Finding Jordan Canonical form given the minimal and characteristic polynomial.

I have the following information: the characteristic polynomial of $A$ is $p_A(t)=(t-4)^3(t+6)^2$ and the minimal polynomial is $q_A(t)=(t-4)^2(t+6).$ I'm having problems seeing how one would work ...
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Obtaining the change of basis matrix to the Jordan matrix

Introduction and description of my problem I have trouble when finding the matrix change of base $P$ that allows me to obtain the Jordan form from the matrix $A$, in other words, find $P$ that ...
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Is there a way to calculate exponent $n$ in matrix vector product: $w=M^nv$

Find $n$ for given square matrix $M$ and vectors $v,w$ in $$w=M^nv$$ Trial (updated) (as vujazzman suggested) Jordan normal form: $$ w = (A J^n A^{-1})v$$ $$ A^{-1}w = J^n A^{-1}v$$ After this ...
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Proof connecting Jordan canonical basis and cycles of generalized eigenvectors

Let T be an operator on a finite dimensional vector space V. Let B be an ordered basis for V. Prove that B is a Jordan canonical basis if and only if B is the disjoint union of cycles of generalized ...
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Proof that SN (or Jordan-Chevalley) Decomposition is unique?

Let $M$ be a matrix with entries in $\mathbb C$. The SN (or Jordan-Chevalley) decomposition theorem states that we can find unique matrices $S$ and $N$ such that: $M=S+N$ $S$ is diagonalizable $N$ is ...
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Eigenvalues of matrix exponential and its Jordan form

Given a matrix $A$, we can write the Jordan decomposition as $$A=SJS^{-1}$$ My question is whether the followings now holds: $$\text{eig}(e^{At})=\text{eig}(e^{Jt})$$ I've tried relating the ...
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Degree of (x-λ) in minimal polynomial

λ is a root of p(x), the minimal polynomial of T (linear operator on complex V). Then λ is an eigenvalue of T. How to prove that the degree of (x-λ) in p(x) equals the size of the largest λ-Jordan ...
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Linear maps satisfying $T^2 = I_n$ and Jordan Form of $T(X) = AX - XA$

Let $V$ be any $n$-dimensional vector space and $W = M_2(\mathbb{C})$. (a) Construct all linear maps $T: V \to V$ such that $T^2 = I_n$. For $T^2 = I_n$, taking $p(X) = X^2 - 1$ we must have $p(...
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Jordan Normal Form of $A_{\pi}: \mathbb{C}^n \to \mathbb{C}^n$ given by $A_{\pi}(v) = A_{\pi}(v_1,…,v_n) = (v_{\pi(1)},…,v_{\pi(n)})$.

Let $\pi \in S_n$ be a permutation. Prove that $A_{\pi}: \mathbb{C}^n \to \mathbb{C}^n$ given by $A_{\pi}(v) = A_{\pi}(v_1,...,v_n) = (v_{\pi(1)},...,v_{\pi(n)})$. Show that $A_{\pi}$ is linear and ...
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Is this a family of similar matrices $\left(\begin{smallmatrix} 0&x\\ 0&0 \end{smallmatrix}\right)$?

Is matrix $A = \begin{pmatrix} 0&1\\ 0&0 \end{pmatrix}$ similar to matrix $B=\begin{pmatrix} 0&2\\ 0&0 \end{pmatrix}$? If so, how do I prove this? I came here from following the ...
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Matrix exponential using Jordan form

I'm trying to calculate the matrix exponential $e^{At}$ for $$A=\frac{1}{2}\begin{bmatrix}-1&1&-1\\2&-2&0\\1&-1&-1\end{bmatrix}$$ I found the eigenvalues $\lambda_1=\lambda_2=-...
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Finding the Jordan canonical form when the characteristic polynomial does not split?

A problem on the 2009 qualifying exam for Harvard is the following: Suppose $\phi$ is an endomorphism of a 10-dimensional vector space over $\mathbb{Q}$ with the following properties: The ...
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Understanding the proof of Jordan normal form, by matrix trick

I read a proof for JNF which I am unclear. Proof: For any complex matrix $A$. Assume $Av_1=\lambda v_1$ for some $v_1 \in \mathbb{C}^n$, then $A(v_1, \cdots, v_n)=(v_1, \cdots, v_n)\begin{pmatrix}\...
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Show exists a subspace $W \subseteq \mathbb{C}^{n}$ of dimension $1$ such that every Jordan basis of $ \mathbb{C}^{n}$ contains a generator of $W$

Let $n\geq 2$. Given $f$ nilpotent endomorphism of $\mathbb{C}^{n}$ such that exists an integer $k \geq 1$ such that $dim \hspace{0.1cm} Kerf^{k+1} = dim \hspace{0.1cm} Kerf^{k}+1$. $(1) \hspace{0....
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Find a Jordan basis for the linear operator $T$

Find a possible Jordan basis for the linear operator $T$ such that: $T(x, y, z, t) = (2y, −2x + 4y, z + t, z + t)$ Is there an specific method to find a Jordan basis? Since I'm teaching myself I'm ...
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Finding the Jordan Canonical Form of a Classical Adjoint of a Jordan Block

Let $A$ be a size $n$ Jordan matrix with $0$ on its diagonal, that is $$A = J_n(0) = [a_{ij}] = \begin{cases} 1, &j=i+1\\ 0, &\text{elsewhere} \end{cases} $$ What is the Jordan Canonical ...
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System of equations involving complex eigenvalues

Consider the following equation: $$ x_{n+2}-2ax_{n+1}+x_n=0$$ a) Define a new auxiliary variable $y_n = x_{n+1}$ and rewrite the previous equation as a discrete, two-equation dynamical system. b) What ...
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Finding the Jordan form given nullities

The question: "Let $B$ be a $10 \times 10$ matrix and let $\lambda$ be a scalar. Suppose it is known that $$ \text{nullity}(B - \lambda I) = 5, \\ \text{nullity}(B - \lambda I)^2 = 8, \\ \text{...
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Jordan canonical form with 2 is the dimension of eigenespace,

$$B= \begin{pmatrix} 0 & 1 & 0 \\ -4 & 4 & 0 \\ -2 & 1 & 2 \end{pmatrix}$$ I need to find Jordan decomposition of B. My sketch: We find Jordan's matrix ...
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Similarity class of $3 \times 3$ matrices with entries in $\mathbb{F}_3$

I've been trying to solve the following problem. Find a representative for each similarity class of $3 \times 3$ matrices $A$ with entries in $\mathbb{F}_3 = \mathbb{Z}/3\mathbb{Z}$ such that $A^4 = ...
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Show The Jordan Normal Form Of $\varphi$.

Fix a nonnegative integer $n$, and consider the linear space $$\mathbb{R}_n\left [x,y \right ] := \left\{ \sum_{\substack{ i_1,i_2;\\ i_1+i_2\leq n}}a_{i_1i_2}x^{i_1}y^{i_2}\quad\Big|{}_{\quad}a_{...
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Can't find the Jordan form of this 3x3

I have the matrix $$\begin{pmatrix} 2 & 2 & -1 \\ -1 & -1 & 1 \\ -1 & -2 & 2 \end{pmatrix}$$ and need to find its Jordan canonical form. I can find that the only eigenvalue ...
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Matrix exponential for Jordan canonical form

Let $X$ be a real $n \times n$ matrix, then there is a Jordan decomposition such that $X = D+N$ where $D$ is diagonalisable and $N$ is nilpotent. Then, I was wondering whether the following is ...
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Jordan Normal form as consequence of the Classification theorem for finitely generated modules over PID

Let $V$ be a $n$-dimensional $\mathbb{C}$-vector space, so $V\cong \mathbb{C}^n$. Let further $T:\mathbb{C}\to \mathbb{C}$ be a $\mathbb{C}$-linear transformation. We consider $V$ as a $\mathbb{C}[X]$ ...
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Finding $\bf{P}$ such that $\bf{P^{-1}AP=B}$ for both fixed $\bf{A},\bf{B}$.

How can I find a matrix $\bf{P}\in \mathbb{{R}^{n\times n}}$, such that $\bf{P^{-1}AP=B}$,where $$\bf{A}=\begin{bmatrix} \bf{A_2}& \bf{C_2}& \\ & \bf{A_2}& \bf{C_2}& \\ ...
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Number of Orthogonal Matrices over R in Jordan normal form [closed]

Is there any way to find the number of Orthogonal Matrices over the real field in Jordan normal form?
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Possible Jordan Canonical Forms: Intuition

As I was reviewing linear algebra before I head off to grad school in the fall, I came across a question about Jordan Canonical Forms. It reads: "Suppose that A is a square complex matrix with ...
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Can a matrix be similar to a block matrix with Jordan Block or companion matrix of the non-linear irreducible factors in its diagonal block?

Let $A$ be $3 \times 3$ real matrix with minimal polynomial $f(X)=(X-1)(X^2 +1)=X^3-X^2+X-1.$ Then By Rational Canonical Form we know that $A$ is similar to the Companion matrix of $f(X)$ which is $\...
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Jordan form of operator $X \mapsto AXA$ [closed]

Matrices $n \times n$ on complex field. Compute Jordan form of operator $X \mapsto AXA$: $$ A = \begin{bmatrix} 0 & 1 & & \\ & 0 & \ddots & \\ & & \...
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Analogy of Jordon Normal Form for Antilinear Maps

Given complex vector spaces $V$, and antilinear $T:V \rightarrow V$, then if we fix a basis of $V$, we can represent $T$ by the matrix of the linear $T \circ J$, where $J$ is complex conjugation. I ...
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When can we bring a matrix to its Jordan form within a subfield of $\mathbb C$?

Can the following matrix $A$ be brought into Jordan form over the field of rational numbers? $$ A=\begin{pmatrix}-3&-1&-1\\6&4&1\\6&5&0\end{pmatrix} $$ My solution: By ...