# 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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### $V/\ker(T-5I)$ is nilpotent.

A linear operator $T$ on a complex vector space $V$ has characteristic polynomial $x^3(x-5)^2$ and minimal polynomial $x^3(x-5)$. Choose correct options. The operator induced by $T$ on quotient ...
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### Prove: $\det(I+A) = 2^{\text{rank}(A)}$ if $A$ is a square idempotent matrix. Find $(I+A)^{-1}$ such that the expression doesn't have inverses.

To prove: $\det(I+A)$ = $2^{\operatorname{rank}(A)}$ if $A \in$ $\mathbb{R}^{n\times n}$ and $A^{2}=A$. Find an expression for $(I+A)^{-1}$ such that it does not involve inverses. Is there any way I ...
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### Jordan decomposition of Idempotent matrix.

Matrix A $\in$ $\mathbb{R}^{n\times n}$ is idempotent if $A^{2} = A$. Describe the Jordan form of A. How do I do this? I am able to decompose a matrix to its Jordan form given that the matrix contains ...
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### Prove than a square matrix $A$, with complex entries, is diagonalizable if and only if the minimal polynomial of $A$ has distinct roots.

Question: Prove than a square matrix $A$, with complex entries, is diagonalizable if and only if the minimal polynomial of $A$ has distinct roots. In this answer Prove that T is diagonalizable if and ...
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### Let $A$ be nilpotent such that $A^{n-1}\neq 0$. Show that $A$ has exactly one Jordan Block.

Question: Let $A$ be $n\times n$ and nilpotent such that $A^{n-1}\neq 0$. Show that $A$ has exactly one Jordan Block. My Attempt: Since $A$ is nilpotent, $A^k=0$ some $1\leq k\leq n$. So, the ...
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### Determining the transformation matrix for Jordan normal form.

Let $$A= \begin{pmatrix} 2 & 0 & -4 & -4 \\ 0 &4&2&3\\ 2&0&8&4\\ -1&0&-2&2\\ \end{pmatrix}$$ I want to find Jordan normal form of $A$ and the ...
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### About Jordan normal form and stability in GIT

I am preparing linear algebra exam and I met a problem about Jordan normal form. Suppose $V$ is a vector space over field $\mathbb K$, $\psi$ is a linear transform on $V$, the char polynomial and ...
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### Form of Jordan block

I'm doing an exercice about Jordan matrix, and I have to write the Jordan matrix of : A =\begin{pmatrix} 3 & 0 & 1& 0&0 &0 &0\\ 0 & 3 & 0& 1& 0&0 &0\\ ...
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### Number of Jordan boxes of size j for eigenvalue $\lambda$ is $2\dim\ker(A-\lambda I)^j-\dim\ker(A-\lambda I)^{j+1}-\dim \ker(A-\lambda I)^{j-1}$ [duplicate]

My question is about the computation of the Jordan normal form. Let $K$ be field. Let $A \in K^{n \times n}$ be a matrix, that characteristic polynomial splits into linear factors over $K$. I've ...
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### Jordan form of 3 x 3 repeated eigenvalue

Consider the matrix \begin{pmatrix} 1 & -3 &1 \\ 1 & 5 & -1 \\ \ 2 & 6 &0 \\ \end{pmatrix} This has eigenvalue 2 with 3 multiplicity....