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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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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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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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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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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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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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 ...
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Why is it not sufficient that geometric multiplicity is equal to algebraic multiplicity to imply that $A$ diagonalizable

I have been told that for a given matrix $A$: $A \operatorname{diagonalizable} \Rightarrow m_{a}(\lambda)=m_{g}(\lambda)$ for all $\lambda \in \sigma (A)$ where $\sigma (A)$ denotes the spectrum of $A$...
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Kronecker-Weierstrass problem, 3x6 matrices conjugacy or congruent classes?

I'm here again with a somewhat vague and hard question our teacher asked us, we have to check and proof that for all matrices $A$ and $B$ that by applying certain simultaneous transformations, we can ...
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Non-nilpotent and non-invertible matrices that have the same characteristic and minimal polynomials have the same Jordan-form

I've come across the following question and am not sure why the answer makes sense. Let $f,g \in End(\mathbb{C}^4)$ be neither nilpotent nor invertible with their characteristic and minimal ...
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Relation between matrix power and Jordan normal form

(a) Assume $A\in\mathbb{C}^{n\times n}$ has $n$ distinct eigenvalues. Prove that there are exactly $2^n$ distinct matrices $B$ such that $B^2 = A$ (i.e., in particular, there are no more than $2^n$...
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Find the permutation matrix

Let: $$J=\begin{bmatrix} \lambda&1&0\\ 0&\lambda&1\\ 0&0&\lambda \end{bmatrix}$$ Find a permutation matrix $M$ such that $$M J M^{-1} = J^{t}$$ I know that $J$ is a ...
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Jordan Block of a complex matrix, with $A^4=I$

The following statement is false or true: If $A \in M(n, \mathbb{C})$ is a matrix with complex entries of order $n$ such that $A^4=I$ then \begin{pmatrix} i & 1\\ 0 & i \end{pmatrix} ...
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Jordan Canonical form with zero eigenvalue?

Can anyone tell me how to find the Jordan Canonical form of the matrix below? $$A=\begin{pmatrix} 0 & 1 & 2\\ 0 & 0 & 1\\ 0 & 0 & 0 \end{pmatrix}$$ Obviously this matrix ...
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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 ...