Linked Questions

Let $A = \begin{pmatrix} 1 & 1& 1\\ 1 & 2 &3 \\ 1 &4 & 5 \end{pmatrix}$ and $D = \begin{pmatrix} 2 & 0& 0\\ 0 & 3 &0 \\ 0 &0 & 5 \end{pmatrix}$....

Suppose $A \in \mathcal{M}_{n}(\mathbb{C})$ is an $n$ by $n$ matrix over the field $\mathbb{C}$. Prove that the following two statements are equivalent. The (monic) characteristic polynomial of $A$ ...

I know that matrix multiplication in general is not commutative. So, in general: $A, B \in \mathbb{R}^{n \times n}: A \cdot B \neq B \cdot A$ But for some matrices, this equations holds, e.g. A = ...

Like the title says, can a matrix in $\mathbb{R}$ have a minimal polynomial that has coefficients in $\mathbb{C}$? I have a feeling the answer is yes because the characteristic polynomial can have ...

Let $A,B$ be $n \times n$ matrices over the complex numbers. If $B=p(A)$ where $p(x) \in \mathbb{C}[x]$ then certainly $A,B$ commute. Under which conditions the converse is true? Thanks :-)

This is part $2$ of the question that I am working on. For part $1$, I showed that the space of $5\times 5$ matrices which commute with a given matrix $B$, with the ground field = $\mathbb R$ , is a ...

Let $A$ be a $n\times n$ matrix with characteristic polynomial $$(x-c_{1})^{{d}_{1}}(x-c_{2})^{{d}_{2}}...(x-c_{k})^{{d}_{k}}$$ where $c_{1},c_{2},...,c_{k}$ are distinct. Let $V$ be the space of $n\...

Let A be a matrix $n\times n$, given a n-vector $v$, what conditions over $v$ and $A$ are necessaries for $[v, Av,..., A^{n-1}v]$ will be linearly independent? For example if $v$ is a eigenvector or $...

I am trying to learn some linear algebra, and currently I am having a difficulty time grasping some of the concepts. I have this problem I found that I have no idea how to start. Assume that $\bf A$...

Let $V=M(n,\mathbb C)$. For a subset $S \subseteq V$, let $C(S):=\{A \in V | AB=BA, \forall B \in S \}$ . How to prove that for every $A\in V$, we have $C(C (\{A\})) \subseteq \{ p(A) | p(t) \in \...

There have been many questions in the vein of this one, but I can't find one that answers it specifically. Suppose $A,B\in M_n(\mathbb C)$ are two matrices such that, for any other matrix $C\in M_n(\...

I'm trying to learn linear and abstract algebra on my own and have been attempting textbook exercises and problem sets I find online. I've been doing okay so far but I found this problem and I'm ...

Question is to prove that : characteristic and minimal polynomial of $ \left( \begin{array}{cccc} 0 & 0 & c \\ 1 & 0 & b \\ 0 & 1 & a \end{array} \right) $ is $x^3-ax^2-bx-c$...

I was looking at Gilbert Strang's lectures on Linear Algebra and noticed that in lecture 2, Elimination with Matrices, around the 40nth minute he mentions that you can use the permutation matrix, $$P=...

Prove that the set of real commuting matrices with the matrix $A= \begin{bmatrix} 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \\ ...