Questions tagged [jordan-normal-form]

This tag is for questions relating to the Jordan normal form, also known as a Jordan canonical form or JCF of a matrix which is a similar block matrix having diagonal blocks when the matrix is diagonalizable and diagonal + nilpotent blocks more generally.

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Is the real Jordan form unique?

Let $A\in M_n(\mathbb{R})$. I know that $A$ has a real Jordan form which is obtained from the complex Jordan form by using the usual Jordan blocks for the real eigenvalues and by associating to the ...
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How can I find possible non-symmetric $A$ if $A^k$ is symmetric?

Assume $\bf A\in \mathbb R^{n\times n}$ If I know ${\bf A}^k$ and that it is symmetric, how can I systematically find the $\bf A$ which are not? Own work One approach I have considered is to assume a ...
mathreadler's user avatar
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Find the change of basis matrix so that the following is in Jordan Normal Form

Let the following matrix be given. Note that we are in the field consisting of five integers: F = (0, 1, 2, 3, 4) $A = \begin{bmatrix} 1 & 2 & 0\\ 3 & 2 & 1\\ 0 & 2 & 2 \end{...
Newbie1000's user avatar
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How do I go from the Jordan form to the Weyr form?

I am currently studying material from a text called "Advanced Topics in Linear Algebra" by O'Meara, Clark, and Vinsonhaler. A large topic in the book is the Weyr form of a matrix. The ...
Noel 's user avatar
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If characteristic polynomial of $A$ is $(x-1)^n$, then show $A$ and $A^{-1}$ are similar

I have a problem about matrix similarity: Let $A$ be an $n\times n$ complex matrix whose characteristic polynomial is $(x-1)^n$. I want to show that $A$ and $A^{-1}$ are similar matrices. I know ...
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A $4\times 4$ matrix counterexample. [duplicate]

A question in Dummit & Foote is asking to prove that two $3\times 3$ matrices are similar iff they have the same characteristic and the same minimal polynomial. I was able to prove that. But then ...
Hope's user avatar
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Decomposing a matrix $M$ in the form $M = P^{-1}QP$ where $Q$ and $P$ are real matrices and Q is as diagonal as possible

I am currently working on a tiny matrix library in C++ to help myself learn more about them. So far, I have implemented basic functions such as addition, subtraction, multiplication, the determinant, ...
Om Patil's user avatar
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what are the differences between Jordan Canonical forms and rational canonical forms?

I am studying JCF and RCF from Dummit & Foote. I can say what are all the differences between them, either in computations or in definitions and constituents. Could someone clarify this to me ...
Emptymind's user avatar
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How can an operator be diagonalizable on its nontrivial root subspace?

My question is about the following problem from "Introduction to representation theory" by Etingof et al. Problem 2.15.1 d): Prove that $H$ is diagonalizable on the root subspace $\overline{...
Daigaku no Baku's user avatar
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Confusion about the primary decomposition theorem in linear algebra

I am currently studying the Jordan canonical form which uses the primary decomposition. I have seen the generalised eigenspace decomposition and I know that the algebraic multiplicity which appears in ...
Wintermelon423's user avatar
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given the minimal polynomial of matrix $A^2$ find minimal polynomials and Jordan forms of $A$

I'm refreshing some linear algebra and came across this question: Consider a matrix $A \in \mathbb{C}^{4\times 4}$ and suppose the minimal polynomial of its square is $\phi_{A^2}=(x-1)^2$. Now we're ...
rikdb's user avatar
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finding Jordan canonform of $A^3$ given the minimal polynomial of $A$

I'm having trouble with the following question: we were given the following minimal polynomial of a matrix $A\in \mathbb {C}^{4 \times 4}: X^4-2X^3+X^2$. Now we are asked to find the Jordan canonical ...
rikdb's user avatar
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Effect of matrix multiplication on ordering of absolute angle between vectors

Consider two vectors $\mathbf u,\mathbf v$. Let $\theta_0$ be the angle between them. Now multiply them by some matrix $M$. Let $\theta_1$ be the angle between $M\mathbf u$ and $M\mathbf v$. W.L.O.G ...
jdods's user avatar
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Relation between nilpotent matrix and matrix with simple elementary divisors

I am studying the paper, Goto, Morikuni, On algebraic Lie algebras, J. Math. Soc. Japan 1, 29-45 (1948). ZBL0038.02104. Here the author proves the unique decomposition of a matrix into a nilpotent ...
Monica's user avatar
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What exactly is the Jordan Canonical Form for the following matrix?

Here is the matrix I am trying to find its JCF: $$ \begin{pmatrix} 1 & 2 & 0 & 0\\ 0 & 1 & 2 & 0\\ 0 & 0 & 1 & 2\\ 0 & 0 & 0 & 1 \end{pmatrix} $$ I ...
Brain's user avatar
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Deduce Jordan Normal Form from PID finitely generated module structure.

Let $k$ be an algebraically closed field. Any $k$-linear map $\phi:k^n\to k^n$ imposes an extra $k[x]$-module structure on $k^n$ by defining $p(x)\cdot v = p(\phi)(v).$ Clearly then $k^n$ is a ...
user108580's user avatar
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Show supremum norm of solution to ode is less than $k$ i.e. $||e^{At}|| \leq k$.

So I'm trying to make an exercise in ode's. Let $A \in M_5(\mathbb{R})$ be a matrix with 3 eigen-values $\lambda_1 = -1$ with multiplicity $3$ and $\dim(\text{Ker}(A+I))=1$, $\lambda_{2,3} = \pm 2i$. ...
Dorelanië's user avatar
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All possible block diagonal forms for real valued $4\times 4$ matrices

This is one part of a larger question I'm trying to answer. The first part was determining all the possible Jordan normal forms for complex $4\times 4$ matrices, and the second part was determining ...
pyat's user avatar
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Proof for algorithm of Jordan normal form using cyclic basis

Suppose $A\in M_n(\mathbb R)$ with \begin{align} \big|A-\lambda I_n\big|\,&=\,\pm(\lambda-\lambda_1)^{m_1}\cdots(\lambda-\lambda_p)^{m_p} \newline minpoly_A(\lambda)\,&=\,(\lambda-\lambda_1)^{...
PermQi's user avatar
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2 answers
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Solve the matrix equation for $X$

Solve $X^{7}=\begin{pmatrix} 1 & 0 & 1 \\ 0 & -1 & 0 \\ 0 & 0 & 1 \\ \end{pmatrix}$. We dont know the field where the entries of $X$ are. The matrix $A=X^{7}$ is not ...
Stefan Solomon's user avatar
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Numerical inaccuracy in diagonalizing a matrix / finding its Jordan form

I was trying to numerically compute eigenvalues and eigenvectors of a matrix in Python with Numpy. The problem was that the inverse similarity transformation of the eigenvalue matrix does not ...
pawat akara-pipattana's user avatar
2 votes
1 answer
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Finding all possible Jordan forms from the Characteristic polynomial

Let A be a 7 x 7 matrix with characteristic polynomial $(t − 2)^4(3 − t)^3$. It is known that in the Jordan form of A, the largest blocks for both the eigenvalues are of order 2. Show that there are ...
Ddh Hhd's user avatar
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Is there some geometric understanding of a Jordan normal matrix?

We know that an eigenvalue decomposition of a matrix is to find those eigenvectors that are just scale for some coefficients. But what about Jordan matrix decomposition? I just learn how it is ...
Yuki's user avatar
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Question on a simple proof of Jordan Normal Form

I am currently a year 2 uni mathematician, understanding the Jordan Normal Form in Linear Algebra. Here is a paper for a short proof of the Jordan Normal Form, which I am looking at in order to ...
CatsAndDogs's user avatar
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Suppose $AB=BA$ and $x$ is an eigenvector of $A$ with geometric multiplicity $k$, can $Span\{x, Bx, ...\}$ be represented?

Suppose $A$ and $B$ are $n \times n$ commuting matrix: $AB=BA$. $x$ is an eigenvector of $A$ with the eigenvalue $\lambda$ of geometric multiplicity $k$ i.e. $\ker(\lambda I-A)$ is of dimension $k$. ...
Pina Rinith's user avatar
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Possible non similar jordan Forms

Be A an M_6(R) matrix so that A^4-8A^2 +16I = 0. What are the possible Jordan Forms non similar to A? I tried to solve that, so I arrived in the equation (A-2I)^2*(A+2I)^2 = 0. So this will give me ...
malviska's user avatar
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Understanding this step in finding the transition matrix

There's such a problem about finding the transition matrix. Let $\mathbf A=\begin{bmatrix}2&6&-15\\1&1&-5\\1&2&-6\end{bmatrix}$, find non-singular matrix $\bf P$ such that $\...
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Matrix exponentials of Jordan blocks

Let $f:\Bbb C→\Bbb C$ be an analytic function1. Let $\lambda \in\Bbb C$ and let $$ J=\begin{pmatrix} \lambda & 1 & & &\\ &\ddots& \ddots \\ & & \ddots& 1\\ &...
User1's user avatar
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If $f:\ V\longrightarrow V$ be linear then how the fact $V=ker\,f+im\,f$ relate to the diagonalization / Jordanization of $f$?

Let $V$ be a n-dimensions vector space and $f$ be a linear map on $V$. Then is there any relevant between $V=ker\,f+im\,f$ and the diagonalization/Jordanization of $f$ ? Thanks
PermQi's user avatar
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On Orthogonality of Generalized Eigenspaces

I was reading a paper, and it made a claim that for some nilpotent matrix $A$, we can say that we can find a Jordan basis of $A$ that is orthonormal. I understand that what this means is that all sets ...
Joshua G-F's user avatar
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1 answer
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General expression of the exponential function of a matrix

For matrix $A \in \mathbb{C}^{n\times n},$ it is well-known that there exists an invertible $P$ so that $PA=JP,$ where $J$ is the Jordan canonical form of $A$. Hence we have $e^A=P^{-1}e^JP.$ For real ...
vent de la paix's user avatar
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A question about right stable matrix

Let $n \times n$-matrix A have precisely $k(1\leq k \leq n)$ stable eigenvalues(stable means the real part is strictly negative). A full-rank $n \times k$-matrix $R_A^S$ is called a right stable ...
vent de la paix's user avatar
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Find a linear transformation $f$ on $\mathbb R^4$ for which ker $f = \langle (1,0,1,0) \rangle$ and Im $f^2 = \langle (1,0,1,0), (1,1,1,1) \rangle$

(i) Find a linear transformation $f: \mathbb R^4 \to \mathbb R^4$ for which $$\ker(f) = \langle (1,0,1,0) \rangle \text{ and Im }(f^2) = \langle (1,0,1,0),(1,1,1,1) \rangle.$$ (ii) Find all the ...
Squirrel-Power's user avatar
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Jordan Form of a Unipotent Operator Induced on a Quotient Space

Let $K$ be an algebraically closed field, let $V$ be a finite dimensional $K$-vector space, and let $u:V \rightarrow V$ be a unipotent operator. For each $j \in \mathbb{Z}^+$, let $c_j$ be the number ...
Drew's user avatar
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Can we make the Jordan Normal Form Bicontinuous?

I will be using the set-theoretic notation $0=\emptyset$, $\forall n\in\mathbb{N}^*: n=\{0,1,...,n-1\}$ and $B^A=\{f:A\to B\}$ so the cartesian power $X^n$ can be expressed as $X^n =\{f:n\to X\}$ ...
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if $\phi$ is a linear trasformation on V, its matrix under a group of base vectors is a jordan block, is this V is a cyclic space under $\phi$?

I only know if jordan block is $J_n(0)$ then V will be cyclic space under corresponding linear transformation $\phi$. Will this right for any $J_n(a)\ (a\neq 0)$? I was really confused while ...
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Finding Jordan canonical form on a group algebra

I am preparing for my algebra qualifying exam, and this is one of the past exam problems, which I got stuck on. I would appreciate any help! Let $F=\mathbb{Z}/11\mathbb{Z}$, the field of $11$ elements....
mathlearner98's user avatar
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Given A $\in \Bbb F^{14 \times 14}$ such that $rank (A^4)=0$ $rank (A^3)=1$ $rank (A^2)=4$ $rank (A)=8$ find the matrix jordan nromal form

Given A $\in \Bbb F^{14 \times 14}$ such that $rank (A^4)=0$ $rank (A^3)=1$ $rank (A^2)=4$ $rank (A)=8$ find the matrix Jordan normal form The answer in the textbook is $J_A=diag(J(0,4),J(0,3),J(...
Adamrk's user avatar
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Classification of all similar $3\times 3$ matrices over $\mathbb{R}$

I want to ask a question regarding classes of similar $3 \times 3$ matrices. We were told that two matrices are similar if and only if they have the same Jordan normal form. That led me to trying to ...
watertrainer's user avatar
3 votes
1 answer
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Finding the matrices $J$ (Jordan normal form) and $T$ so that $J=T^{-1}AT$

To solve an automation engineering exercise, I need to find the Jordan normal form $J$ and a matrix $T$ so that $J=T^{-1}AT$, where $A$ is the initial matrix (given by the exercise). For example: $$A =...
marcocipriani01's user avatar
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Finding the Jordan Normal Form of a matrix

Let $ A \in M_{n\times n}\left(\mathbb{C}\right) $, such that $ A^{2008}=O $ and $ 0 \lt 3\rho\left(A^2\right)\lt2\rho\left(A\right)\lt5 $. I need to find its Jordan Normal Form. Did I do it correctly?...
talopl's user avatar
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How to calculate the Jordan canonical form of the following matrix?

This is an exercise from my textbook. "Calculate the Jordan normal form of the following matrix: $$A=\begin{pmatrix} 1&2&3&4&\cdots&n\\ 0&1&2&3&\cdots&n-1\\...
PauseAndPonder's user avatar
2 votes
1 answer
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How to compute quickly the Jordan normal form of a product of the form $B A \operatorname{adj} B$? [duplicate]

Compute the Jordan normal form of the product $$BA\operatorname{adj}(B),$$ where $$A = \begin{pmatrix} 1 & 3 & -1 \\ -1 & 4 & 0 \\ 0 & -1 & 4 \\ \end{pmatrix}, \qquad B = \...
Tymofii256's user avatar
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Is there an example that a specific matrix has Jordan block $J_{2}(i)$?

Consider a matrix of the form $\begin{bmatrix} A & C\\ -C^{T} & B \end{bmatrix}$ where A and B are symmetric matrices. Can matrices of this type have a Jordan normal form representation ...
Kyuwon Kim's user avatar
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$3\times 3$ Matrix exponent [duplicate]

I have a problem where I’m given the matrix $$ B = \begin{pmatrix} 1 & 1 & 0\\ 0 & 1 & 1\\ 0 & 0 & 1 \end{pmatrix}. $$ I’m tasked with computing $e^B$. Now the point where I’m ...
Sebastian Gersmeier's user avatar
2 votes
0 answers
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Finding all invariant subspaces of $T_A:\mathbb{C}^n \to \mathbb{C}^n$ where $A=J_{n_1}(\lambda_1)\oplus ... \oplus J_{n_k}(\lambda_k)$

Let $A=J_{n_1}(\lambda_1)\oplus ... \oplus J_{n_k}(\lambda_k)\in \text{Mat}_n(\mathbb{C})$ when $\lambda_i\neq\lambda_j$ for every $i\neq j$. I need to describe all the $T$-invariant subspaces of $T=...
Staltus's user avatar
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Commuting matrices and their Jordan forms

I am recently studying commuting matrices. I was reading the book Invariant Subspaces of Matrices with Applications(Godberg, Lancaster and Rodman) and pg.295-296 claims that a matrix $X$ is a solution ...
Ahmet Sakal's user avatar
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Proving that every invariant subspace of $\mathbb{F}^n$ is of the form $span(e_1,...,e_k)$

Let $T_{J_n(0)}:\mathbb{F}^n \to \mathbb{F}^n$. I need to prove that every invariant subspace $W$ of $\mathbb{F}^n$ is of the form $Span(e_1,...,e_k)$ for some $0\leq k \leq n$. This is what I've got ...
Staltus's user avatar
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minimal polynomial of finite field

$\frac{d}{d Y}: K[Y] \rightarrow K[Y], \quad \sum_{i \geq 0} a_{i} Y^{i} \mapsto \sum_{i \geq 0}(i+1) a_{i+1} Y^{i}$ $P_n =\[f \in K[Y] \mid \operatorname{deg}(f) \leq n]$ (this is a set) $f=\left. \...
Marius Lutter's user avatar
2 votes
1 answer
113 views

Order of eigenvectors within basis for Jordan Normal Form?

I'm currently baffled as I thought that the order of eigenvectors within the basis of a JNF decomposition doesn't matter. I may have a made a mistake in my working, but if not, is there a general rule ...
Dillon Shah's user avatar

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