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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Using power of matrix to find JCF

Given a $5 \times 5$ matrix $A$, find any Jordan canonical form for $A$. There is a hint, that you should calculate $A^3$ first (crucially, not $(A-\lambda E_5)^3$). What information does the power ...
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If $B$ is nilpotent and $AB=BA$ then $\det(A+B)=\det(A)$ (Asking for other method) [duplicate]

Let $K$ be some field and $A, B \in M_n(K)$. Prove that: If $B$ is nilpotent and $AB=BA$ then $\det(A+B)=\det(A)$. I believe there is a nice solution here. However, it seems that this problem could ...
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Let $A, B, C$ be some complex matrices. If $AB - BA = C$ and $AC = CA$, then $C^k = 0$ for some $k$. [duplicate]

Let $A, B, C$ be some complex matrices. Suppose that $AB - BA = C$ and $AC = CA$. Prove that: $C^k = 0$ for some $k$. It is an exercise in the section of "Jordan Canonical Form of Nilpotent ...
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Jordan normal form and spectral decomposition

In this post by Terence Tao (exercise 28, point vi), he proves the following theorem (called spectral decomposition): Theorem: Let $A$ be a complex square matrix. Then $A$ can be written as $A=\sum ...
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Galois descent for a semisimple automorphism

Let $K$ be a perfect field and $\overline{K}$ be the algebraic/separable closure. Let $V$ be a finite dimensional $K$-vector space, and let $V_{\overline{K}} = V \otimes_K \overline{K}$. Given an ...
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Jordan Normal Form Question

I'm trying to properly get to know the Jordan normal form Theorem, and am confused as to why this proposition holds. I have read that if A is a matrix in Jordan normal form and $T:V\rightarrow V$ then ...
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Can we reduce finding matrix roots to finding roots of Jordan blocks?

I just found some interesting question about matrix square roots and I came to think of one way to find them, or at least reduce them to a set of simpler problems. Assume we have a matrix $\bf A$ and ...
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Jordan basis of the matrix with the only one eigenvalue 0

Find the Jordan basis of the following matrix: $$\begin{pmatrix} 1& 1& -2& 3& -1\\ 0 & 0 &-1 &2 &0\\ 2& 2& -6& 10& -2\\ 1& 1& -3& 5& -...
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85 views

Why is Jordan Normal Form unique?

I know that the Jordan Normal Form of a matrix is unique (up to reordering the Jordan blocks), but I don't really see why. Say we're looking at a 3x3 case. Now, all we need to do to compute the Jordan ...
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Given the Jordan form $J$, find matrix $P$

In a question set in my linear algebra course, I'm asked the following: Find $P$ such that $P^{-1}AP=J$, where $$A = \begin{pmatrix}6&5&-2&-3\\ -3&-1&3&3\\ 2&1&-2&...
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Find Jordan norm form of matrix using lambda-matrix techniques

I can't transform this matrix of lambda-matrix techniques. I don't know this method, but I should use this method for solve. Eigenvalues equal to "-1". Guys, help me, please. $$A=\left(\begin{array}{...
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I want to find an explicit direct sum decomposition of the space into T-cyclic subspaces

I construct a linear operator $T \in \mathcal{L(\mathbf{C}^7)}$ , where the minimal polynomial is $m_T (x) = x^2(x-1)^2$ and the caraterisitic polynomial is $p_T (x) = x^3(x - 1)^4$. The linear ...
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Find Jordan Decomposition of $\begin{pmatrix} 4 & 0 & 1 \\ 0 & 1 & 0 \\ 1 & 0 & 3 \end{pmatrix}$ over $\mathbb{F}_5$

Find the Jordan decomposition of $$ A := \begin{pmatrix} 4 & 0 & 1 \\ 0 & 1 & 0 \\ 1 & 0 & 3 \end{pmatrix} \in M_3(\mathbb{F}_5), $$ where $\mathbb{F}_5$ is the field ...
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Show that the characteristic polynomial is the same as the minimal polynomial

Let $$A =\begin{pmatrix}0 & 0 & c \\1 & 0 & b \\ 0& 1 & a\end{pmatrix}$$ Show that the characteristic and minimal polynomials of $A$ are the same. I have already computated ...
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Let $A$ be an $n \times n$ matrix with real entries such that $A^2 + I = 0$ then $n$ is even. Not duplicated

Let $A$ be an $n \times n$ matrix with real entries such that $A^2 + I = 0$ then $n$ is even. And if $n = 2k$, then $A$ is similar over the field of real numbers to a matrix of the block form $$\...
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Canonical Jordan form contradiction

I am faced with the following problem: Given endomorphism $f$ whose characteristic polynomial is $$P_c(x) = (x+1)^{10} (x-1)^{10} x^{10}$$ and whose minimal polynomial is $$P_m (x) = (x+1)^5 (x-1)^...
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let J(A) be the Jordan form of A. and let f be some polynomial. is it true that $\det(xI-f(A))=\det(xI-f(J(A))$ [closed]

I tried a couple of examples and it turned out to be true, but I couldn't prove it..
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Convergence of powers of matrices in Jordan Canonical Form (Jordan Normal Form)

I've actually been stuck on this for a bit while studying for an exam, so would appreciate any help. The problem involves testing whether $\lim\limits_{m \to \infty}$ $A^m$ exists. From lecture ...
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Finding possible Jordan forms of a matrix

Find the Jordan forms of a matrix $A$ subject to the following conditions: the characteristic polynomial is $(x-1)^4(x+3)^5$. matrix $A-I$ has nullity $4$ and matrix $A+3I$ has nullity $1$. ...
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How to find a Jordan basis and a Jordan matrix for a nilpotent matrix?

I am trying to find a general step-by-step "easy" / "intuitive" solution to finding Jordan basis and Jordan matrix (based on the basis) for a nilpotent matrix. If you can add an intuition for the ...
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52 views

How to prove that the following matrices in $M_p(\Bbb F_p)$ is similar

How to prove that the following matrices in $M_p(\Bbb F_p)$ is similar: Consider two matrices $$(a_{ij})= \begin{pmatrix} 1 & 1 & 0 & \cdots & 0 & 0 \\ 0 & 1 &...
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How can I find the Jordan form of this upper triangular Toeplitz matrix?

Given an $n \times n$ matrix $A$ whose $(i,j)$ entry is $$a_{ij} = \begin{cases} n-j+i & \text{if } j \geq i\\ 0 & \text{otherwise}\end{cases}$$ find its Jordan form. I know that ...
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Is this a correct solution? Jordan form of a marix

I am given a matrix $A$ that is defined the following way: the element in row $i$ and column $j$ is $\alpha$ if $i=j$, $1$ if $j=i+2$ and $0$ otherwise. I need to find Jordan Form. This is my ...
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If $V_j = \ker(A-\lambda I)^j$, Why does $\dim(V_j)=r_i$ if and only if $V_j = V_{j+1}$?

If we have a matrix $A$ and the characterisic polynomial is $(x-\lambda_1)^{r_1} \cdot...\cdot(x-\lambda_m)^{r_m}$ and we define for each $\lambda_i$, $V_{j} = \ker(A-\lambda I)^j$. Why does there ...
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Ordering eigenvector basis to produce diagonal matrix J

In order to solve the system \begin{equation} \begin{pmatrix} y_{1}'\\ y_{2}'\\ y_{3}' \end{pmatrix}= \begin{pmatrix} 1 &-1&0 &\\1&3&0\\-2&1&-1 \end{pmatrix}\begin{...
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Can I show that for any matrix A = ST, S & T Both symmetric and invertible matrices

In my linear algebra class, my professor gave us an exercise, which is the following. If A is an nxn matrix over the complex, one can show that A = ST for S and T both invertible and symmetric. I ...
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How to find the invariant factors of a matrix given its Jordan canonical form?

Suppose we have a matrix $A \in M_{12}(\mathbb{C})$ whose Jordan canonical form is $$J_1(0) \oplus J_1(0) \oplus J_2(0) \oplus J_2(0) \oplus J_1(\sqrt{2}) \oplus J_1(\sqrt{2}) \oplus J_1(-\sqrt{2}) ...
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The exponential of a Jordan block

Is it true that the exponential of a Jordan block is an upper triangular matrix? I tried two examples and got just diagonal matrices which may be a coincidence, as diagonal matrices are also upper/...
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Jordan normal as transformation with respect to the basis of eigenvectors

I have the following matrix $$A = \begin{pmatrix} 2 & 0 & 1 & -3 \\ 0 & 2 & 10 & 4 \\ 0 & 0 & 2 & 0 \\ 0 & 0 & 0 & 3 \\ ...
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Hoffman and Kunze ,Linear algebra Sec 7.4 exercise 4

Construct a linear operator $T$ with minimal polynomial $ x^2(x-1)^2 $ and characteristic polynomial $x^3(x-1)^4$. Describe the primary decomposition of the vector space under $T$ and find the ...
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Find the Jordan Canonical Form that is similar with the idempotent matrix A

Find the Jordan Canonical Form that is similar to the idempotent matrix $A$. I know that since $A=A^2$ then $A(A-I)=0$ so the minimal polynomial is $m_A(\lambda)=\lambda(\lambda-1)$. I also know ...
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Proof involving the spectral radius and the Jordan canonical form

Let $A$ be a square matrix. Show that if $$\lim_{n \to \infty} A^{n} = 0$$ then $\rho(A) < 1$, where $\rho(A)$ denotes the spectral radius of $A$. Hint: Use the Jordan canonical form. I am ...
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all 2 dimensional invariant subspaces

How we can find all 2 dimensional invariant subspaces of \begin{pmatrix} 2 & 1 & 0 \\ 0 & 2 & 0 \\ 0 & 0 & 8 \end{pmatrix} I know that there are at least 2 such subspaces, ...
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Computing the matrix exponential for a Jordan matrix

How can I compute $e^{At}$ where $A = J_{3}(5)$? That is, $$A = \begin{pmatrix} 5 & 1 & 0 \\ 0 & 5 & 1 \\ 0 & 0 & 5 \end{pmatrix} $$ Using this, how can I write down a basis ...
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Given a Jordan canonical basis, how to find out to which generalized eigenspace picked generalized eigenvector belongs

Suppose we have finite-dimensional linear operator $A:V\to V$ , that has eigenvalues $\lambda_1 ,\lambda_2, ... \lambda_n$ . It is known that we can decompose $V$ into direct sum of generalized ...
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$Ch_A =(x+1)^6(x-2)^3 $ y $min_A = (x+1)^3(x-2)^2 $, List the possible Jordan forms for $A$

Let $A$ be a complex matrix such that $Ch_A =(x+1)^6(x-2)^3 $ y $min_A = (x+1)^3(x-2)^2 $, List the possible Jordan forms for $A$. And in each case write the corresponding rational I do not know ...
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Deducing the additive Jordan decomposition

In $M_n(\Bbb C)$, I could prove that the additive Jordan decomposition of $X=D+N$ with $D$ diagonalizable and $N$ nilpotent gives a multiplicative Jordan decomposition $e^X=e^De^N$. Is that true the ...
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When $A^i=A^j$ for $i,j\geq 0$ such that $i\neq j$ for matrix $A$ over algebraic closed field?

Let $A\in M_n(K)$ be a matrix over algebraic closed field $K$, where $n>1.$ When $A^i=A^j$ for $i,j\geq 0$ such that $i\neq j$? I tried solve it by Jordan form of matrix $A;$ is is sufficient ...
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Converting Jordan Normal Form into Real Jordan Form

Given the matrix $$\begin{bmatrix} 0 & 0 & 0 & -8\\ 1 & 0 & 0& 16 \\ 0 & 1 & 0 & -14 \\ 0 & 0 & 1 & 6 \\ \end{bmatrix}$$ ...
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Is Jordan cardinal form the matrix having the most zeros in the equivalent class of similarity? [duplicate]

I am concerned on this interesting question Given matrix $A$, does the Jordan cardinal form have the most zeros, in other word, it has the least nonvanishing indices, among the equivalent class of ...
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How do I complete the steps of finding the Jordan of this $5\times 5$ matrix (with Octave)?

I know how to begin the procedure but I don't know how to finish it. Let's start with an example (sorry for it being so unwieldy). Let $$A =\begin{pmatrix} 177& 548& 271& -548& -...
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Generalised Eigenvectors Issue

Ok so i have been doing a few questions on 'Diagonalising' defective matrices, the method I've been using to find generalized Eigenvectors is to make the previous Eigenvector the subject. However i ...
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Decompose an invertible matrix into an exchangeable product of diagonalizable matrix and a matrix with all the eigenvalues equal to $1$

Let $ g $ be an invertible $ n\times n $ complex matrix. Show that $ g $ can be written as $$ g=su=us ,$$ where $ s $ is diagonalizable and all eigenvalues of $ u $ are equal to $ 1 $. My ...
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Can there be just one eigenvalue on the invariant subspace (generalized eigenspace) associated with an eigenvalue?

In the proof of Jordan decomposition here, once I know that an indecomposable subspace $V$ is of the form $V=Ker((f-\lambda Id)^n)$, can there be an other eigenvalue $\mu$ for $f\vert_V$?
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Intuition of generalized eigenvector.

I was trying to get an intuitive grasp about what the the generalized eigenvector intuitively is. I read this nice answer, so I understand that in the basis given by the generalized eigenvectors, a ...
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The theory behind linear recurrence relations solving (or - why does it work?)

tl;dr - a recommendation for a good book that explains the theory behind the auxiliary polynomial/companion matrix methods to solve linear recurrence relations with constant coefficients? I've bumped ...
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60 views

Show that matrix $A$ is similar to a matrix $B$ with elements on diagonal $(0, …, 0, \operatorname{Tr(}A))$ respectively.

Let $A$ be a matrix $n \times n, n \geq 2 $. Let's assume that not all entries outside of the diagonal are zeros (we don't know what entries are on the diagonal). Show that matrix $A$ is similar to a ...
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Let's assume that $ XA = AX $. Show that there is such a matrix $M$ that $ p_A(X) = M(A-X), MA=AM$ and $ MX=XM $.

Let $ A, X \in M_{nxn}(K) $. Let $ p_A(t) $ be a characteristic polynomial of matrix A. Let's assume that $ XA = AX $. Show that there is such a matrix $M$ that $ p_A(X) = M(A-X), MA=AM$ and $ MX=XM $....
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60 views

Exponential of a Jordan Block using Cayley-Hamilton Theorem

For the sake of simplicity, I will only consider $2\times2$ matrices. The Cayley-Hamilton theorem allows us to conclude that $$e^{At} = \alpha_0I + \alpha_1 A$$ where $\alpha_0$ and $\alpha_1$ can ...
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78 views

Find Jordan normal form and basis

Let $$A=M(\varphi)^{st}_{st}={\begin{bmatrix}0&1&1\\-4&-4&-2\\0&0&-2\end{bmatrix}}$$ and $ \varphi: \mathbb R^{3} \rightarrow \mathbb R^{3}$. Find the Jordan normal form $J_{A}$...