# 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.

889 questions
Filter by
Sorted by
Tagged with
17 views

### 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-...
31 views

55 views

### Do there exist two-dimensional subspaces $W$ and $Y$ of $\mathbb{R^4},$ both invariant under $\alpha,$ such that $\mathbb{R^4} = W \oplus Y$?

The question is given below: Could anyone give me a hint for solving the second part of the question?
15 views

### 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, ...
20 views

### 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 $m_i$, ...
51 views

### 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 ...
51 views

15 views

### 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 $5\times 5$ ...
56 views

### 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 $I + N$. ...
31 views

### 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 ...
4k views

### 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 ...
21 views

### 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 ...
46 views

### 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 ...
8 views

### 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 ...
67 views

### 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 ...
51 views

### 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 ...
107 views

### 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 ...
69 views

75 views

### 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 ...
54 views

169 views

### 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 ...
275 views

### 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 ...
40 views

### 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 ...
35 views

90 views

### 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 ...
4k views

### 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 ...
48 views

### 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 $\mathbb{C}[X]$ ...
128 views

How can I find a matrix $\bf{P}\in \mathbb{{R}^{n\times n}}$, such that $\bf{P^{-1}AP=B}$，where $$\bf{A}=\begin{bmatrix} \bf{A_2}& \bf{C_2}& \\ & \bf{A_2}& \bf{C_2}& \\ ... 1answer 33 views ### Number of Orthogonal Matrices over R in Jordan normal form [closed] Is there any way to find the number of Orthogonal Matrices over the real field in Jordan normal form? 1answer 85 views ### Possible Jordan Canonical Forms: Intuition As I was reviewing linear algebra before I head off to grad school in the fall, I came across a question about Jordan Canonical Forms. It reads: "Suppose that A is a square complex matrix with ... 1answer 65 views ### Can a matrix be similar to a block matrix with Jordan Block or companion matrix of the non-linear irreducible factors in its diagonal block? Let A be 3 \times 3 real matrix with minimal polynomial f(X)=(X-1)(X^2 +1)=X^3-X^2+X-1. Then By Rational Canonical Form we know that A is similar to the Companion matrix of f(X) which is \... 1answer 78 views ### Jordan form of operator X \mapsto AXA [closed] Matrices n \times n on complex field. Compute Jordan form of operator X \mapsto AXA:$$ A = \begin{bmatrix} 0 & 1 & & \\ & 0 & \ddots & \\ & & \...
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 ...
### 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 ...