# 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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### Show that a nonzero $2 \times 2$ matrix $A$ such that $A^2 = 0$ is similar to $\begin{pmatrix}0&1\\0&0\end{pmatrix}$

Let $A$ be a $2 \times 2$ non-zero matrix such that $${A}^{2}=0.$$ How do I find an invertible matrix P such that $${P}^{-1}AP=\begin{bmatrix}0&1\\0&0\end{bmatrix} ?$$ Anyone? Please provide ...
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### If $A^2=0$ then $J^2=0$ - Jordan Form

If $A$ over the field $\Bbb C$ and $A^2=0$, we can say that $A$ is similar to a jordan form. Does this mean we are able to say that: $$J^2=0$$ aswell, and why?
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### Jordan form of a matrix

Let $$A = \left( {\matrix{ 0 & 1 & 0 & 0 \cr 0 & 0 & 2 & 0 \cr 0 & 0 & 0 & 3 \cr 0 & 0 & 0 & 0 \cr } } \right)$$ The ...
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### Classify up to similarity all complex $3 \times 3$ matrices A such that A such that ${A^3} = 2{A^2} - A$

Classify up to similarity all complex $3 \times 3$ matrices A such that A such that ${A^3} = 2{A^2} - A$. Here is what I know: All matrices with complex entries have Jordan canonical forms. If ...
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### Identify nilpotent matrix according to its characteristic polynomial (all eigenvalues are $0$)

I was wondering about something. Say $A_{n \times n}$ is a matrix and it's characteristic polynomial is $P(x)=x^n$ (all eigenvalues are $0$), can you say that $A$ is a nilpotent matrix? I really don'...
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### How many Jordan normal forms are there when the characteristic polynomial is $(\lambda+4)^5(\lambda-2)^2$?

Let $A\in M_7(\mathbb{C})$ be a matrix in with the characteristic polynomial $p(A)=(\lambda+4)^5(\lambda-2)^2$. I need to find all Jordan normal forms for this. I think that i can use that the ...
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### Exericse about linear map $T\in L(V)$, where $\dim V=n\geq2$, with $\operatorname{null}T^{n-1}\neq\operatorname{null}T^n$

I have this problem that I am attempting, and am struggling with (b). -- Assume $\dim V = n \geq 2$ and that $T \in L(V)$ such that $\operatorname{null}T^{n-1}\neq\operatorname{null}T{^n}$ -- (a) ...
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### A linear map $T: \mathbb{R^3 \to \mathbb{R^3}}$ has a two dimensional invariant subspace.

Let $T: \mathbb{R^3 \to \mathbb{R^3}}$ be an $\mathbb{R}$-linear map. Then I want to show that $T$ has a $2$ dimensional invariant subspace of $\mathbb{R^3}.$ I considered all possible minimal ...
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### Find $\dim\ker(A-B)$ if the minimal polynomials of $A$ and $B$ are given

Suppose $A,B$ are $3\times 3$ matrices with minimal polynomials $x^2-4$ and $x+2$ resp. What are the possible dimensions of the kernel of $A-B$? The possible JCFs for $A$ are $(2,2,-2), (2,-2,-2)$. ...
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### If matrices $A,B$ similar, find nonsingular $S$ s.t.$B=S^{-1}AS$

Consider the matrices below $$A=\begin{bmatrix}9&4&5\\-4&0&-3\\-6&-4&-2\end{bmatrix}$$ and $$B=\begin{bmatrix}2&1&0\\0&2&0\\0&0&3\end{bmatrix}$$ These ...
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### Can doubly stochastic matrices have non trivial Jordan forms?

This is a followup to a previous question where a nice counter example came up to the proposition "stochastic matrices can only have trivial Jordan forms". This question looks at the more strict case ...
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### Jordan Canonical Form set of polynomials over complex field

N: $P_4(C) \to P_4(C)$ defined by $N(p) = p''- 3p'$ Find the canonical form and a canonical basis for the mapping of $N$. I am not sure how to compute this when it is over $\mathbb{C}$ instead of ...
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### Proving $V=\text{KerT}^n \oplus \text{Im}T^n$ for $V$ in $\mathbb C$

Let $V$ be a vector space over $\mathbb C$ with dimension $n$. Let $T:V\to V$ be a linear transformation. I tried to show that $V=\text{KerT}^n \oplus \text{Im}T^n$ . This is my solution, ...
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### Finding the n-th power of a matrix using Jordan's normal form

There is a matrix $B$ given $$B = \begin{bmatrix} 5&2&-6\\-5&-2&-6\\4&1&-6\end{bmatrix}$$ I am to find $B^n$ using Jordan's normal form. However I have no idea how to deal with ...
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### Finding Jordan form when characteristic is degree 8 and minimal is degree 4.

I'm hoping someone can help me with this question and see if I understand the theory behind it. Suppose we have a linear transformation $T$ represented by a matrix $A$. Let the characteristic ...
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### Jordan Normal Form of $T(X)=AX$

Let $T:M^{F}_{n \times n} \to M^{F}_{n \times n}$ be a linear transformation defined by $T(X)=AX$, where $F$ is a field. The matrix $A$ and the transformation $T$ have the same minimal ...
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### Non-zeroth power of a Jordan block for the eigenvalue $1$ is similar to itself

I'm trying to prove: If $J$ is a single Jordan block corresponding to an eigenvalue $\lambda = 1$, then $J^k$ is similar to $J$, where $k$ is a nonzero integer. Moreover, if $\lambda = 1$ is the only ...
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I have the following matrix $$A = \begin{pmatrix} 2 & 0 & 1 & -3 \\ 0 & 2 & 10 & 4 \\ 0 & 0 & 2 & 0 \\ 0 & 0 & 0 & 3 \\ ... 1answer 56 views ### Let's assume that  XA = AX . Show that there is such a matrix M that  p_A(X) = M(A-X), MA=AM and  MX=XM . Let  A, X \in M_{nxn}(K) . Let  p_A(t)  be a characteristic polynomial of matrix A. Let's assume that  XA = AX . Show that there is such a matrix M that  p_A(X) = M(A-X), MA=AM and  MX=XM .... 1answer 39 views ### A and B be n \times n matrices over the field \mathbb F which have the same characteristic polynomial Lemma: Let N_1 and N_2 be 3 \times 3 nilpotent matrices over field \mathbb F. Then, N_1 and N_2 are similar if and only if they have the same minimal polynomial. Use the result above ... 1answer 116 views ### Jordan canonical form of matrix that A^2=A^t Good morning can someone help me in this exercise. Let A\in M(n,\mathbb{R}) for wich exists an integer k\ge 1 such that A^t=A^k where A^t is the trasposed matrix, let also J(A) the jordan ... 2answers 160 views ### Finding Jordan canonical form of a matrix given the characteristic polynomial I am trying to find the Jordan canonical form of a matrix A given its characteristic polynomial. Suppose A is a complex 5\times 5 matrix with minimal polynomial X^5-X^3. The end goal of the ... 1answer 55 views ### Determine the Jordan normal form of a complex matrix I want to determine the Jordan normal form of a complex matrix A with characteristic polynomial \chi_A(x)=(x+1)^4(x+2)^2, minimal polynomial m_A(x)=(x+1)^2(x+2)^2 and that has the property that ... 1answer 115 views ### How many different structures of Jordan forms in M(n,\mathbb{C})? The possible Jordan forms for n \times n matrices in \mathbb{C} can be found if we know that the characteristic polynomial is:$$ p(x)=(x-\lambda_1)^{k_1}(x-\lambda_2)^{k_2}\cdots (x-\lambda_m)^{...
I have the following square matrix $$A = \begin{bmatrix} 2 & 0 & 0 \\ 6 & -1 & 0 \\ 1 & 3 &-1 \end{bmatrix}$$ I found the eigenvalues: $2$ with algebraic and geometric ...