# 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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### Finding $2$-dimensional invariant subspace

I just want to check that my understanding is correct of invariant subspaces. I was given a matrix A in which I have found that it is invertible, so I know that a $2$-dimensional invariant subspace ...
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### What do you call the converse of an invariant subspace of an operator?

Question. I am looking for the concept converse to invariance: what do we call a set $W$, such that $$T(w) \in W \implies w \in W ?\tag{1}$$ I feel there was a word for this, but I can't recall it, ...
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### Finding Jordan canonical basis from Jordan canonical form for 4x4 matrix [duplicate]

The matrix in standard basis: $$A =\begin{bmatrix}-3&1&3&3\\-10&2&9&9\\-4&0&5&4\\2&1&-3&-2\end{bmatrix}$$ characteristic polynomial is $(λ−1)^3 (λ+1)$ ...
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### A basis that makes a matrix triangular.

Find a basis for $\mathbb C^3$ so that the following matrix is in triangular form: \begin{pmatrix} 0 & 1 & 0 \\ 0 & 0 & 1 \\ 1 & 0 & 0 \end{pmatrix} What are the eigenvalues? I ...
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1 vote
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### $5 \times 5$ nilpotent matrices with the same minimal polynomial and nullity must be similar.

Problem. Suppose $A, B$ are both $5 \times 5$ nilpotent complex matrices with the same nullity and the same minimal polynomial. Prove that $A$ and $B$ are similar. My Question. Is there a 'clever' ...
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### Random nilpotent and upper triangular matrix question

Suppose a $4$ by $4$ matrix $A$ is nilpotent and upper triangular, and all $(i, j)$ entries for $i < j$ are chosen randomly and uniformly in the interval $[−1, 1]$. What are the probabilities that ...
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### How can I guarantee that all the number of generalized eigenvectors are equal to the algebraic multiplicity?

I have found many questions in the Stack but, I couldn't find what I want... Since my major is physics, I'm not good at the terms in mathematics. So it was hard to reach understanding Jordan normal ...
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### Computing the Jordan form of matrix - Getting the null matrix?

I am trying to diagonalise the following matrix $$A = \begin{pmatrix} \sqrt{8} & 0 & 0 \\ 4 & \sqrt{8} & -4 \\ 0 & 0 & \sqrt{8} \end{pmatrix}$$ finding its eigenvalues. The ...
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### How can I "recreate" a nilpotent operator given a Jordan matrix with just zeroes on diagonal?

I have been given a problem where for an algebraically closed field $\mathbb{F}$, a vector space $\dim{V}=n\in\mathbb{N}^+$ , $$0 <n_1<n_2<\ldots<n_{k-1},n_k=n$$ A sequence of integers, ...
1 vote
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### Please tell me good linear algebra books which use the elementary divisors to prove the Jordan normal form theorem.

I want to read linear algebra books which use the elementary divisors to prove the Jordan normal form theorem. Please tell me good linear algebra books which use the elementary divisors to prove the ...
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### Finding Geometric Multiplicity and Size of Jordan block of Eigen Value

Let A be square matrix such that $$|A-xI|=x^4×(x-1)^2×(x-2)^3$$ If $rank(A^3)=rank(A^4)<rank(A^2)$ then geometric multiplicity of Eigen value 0 is? I found that rank of A = 9 - dim eigenspace(0). ...
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### Find all possible Jordan forms with a given minimal polynomial

Let $f(x) = (x-5)^2(x^2-1)$. Find all possible Jordan forms for $7 \times 7$ matrices over $\mathbb{C}$ consiting of 5 elementary Jordan blocks whose minimal polynomial is $f(x)$. I found three ...
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### On the complex matrix equation $AX-XA=B$
I want to show that there exists solution to the matrix equation $AX-XA=B$ if and only if $$\begin{pmatrix} A&0\\ 0&A \end{pmatrix}, \begin{pmatrix} A&B\\ 0&A \end{pmatrix}$$ are ...