Articles of linear algebra

Can a matrix in $\mathbb{R}$ have a minimal polynomial that has coefficients in $\mathbb{C}$?

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 coefficients in $\mathbb{C}$ and so I don’t see any reason why not, I was merely asking in case there was some sneaky proof […]

$AB=BA$ implies $AB^T=B^TA$.

I am looking for an elementary proof (if such exists) of the following: $$ AB=BA \quad\Longrightarrow\quad AB^T=B^TA, $$ where $A$ and $B$ are $n\times n$ real matrices, and $A$ is a normal matrix, i.e., $AA^T=A^TA$ – it is true for complex matrices as well, with $A^T$ replaced by $A^*$. There is a non-elementary proof of […]

How to diagonalize this matrix…

Can someone show me step-by-step how to diagonalize this matrix? I’m trying to teach myself differential equations + linear algebra, but I’m stumped on how to do this. I’d really appreciate if someone would take the time to do this with me! $\begin{bmatrix} 1 & 2 & 0 \\ 2 & 1 & 0 \\ […]

A certain unique rotation matrix

One can find that the matrix $A=\begin{bmatrix} -\dfrac{1}{3} & \dfrac{2}{3} & \dfrac{2}{3} \\ \dfrac{2}{3} & -\dfrac{1}{3} & \dfrac{2}{3} \\ \dfrac{2}{3} & \dfrac{2}{3} & -\dfrac{1}{3} \\ \end{bmatrix} $ is at the same time $3D$ rotation matrix and for it the sum of entries in every column (row) is constant (here $-\dfrac{1}{3}+ \dfrac{2}{3} + \dfrac{2}{3} = 1)$. […]

Condition for commuting matrices

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 🙂

Nth roots of square matrices

Is there a general method (which can be implemented by hand) to finding the $n$-th roots of $2 \times 2$ matrices? Is there a similar method for a general $m \times m$ matrix? (for $n > 1$ and $n\in\mathbb{Z}$)

Intuitive explanation of a positive-semidefinite matrix

What is an intuitive explanation of a positive-semidefinite matrix? Or a simple example which gives more intuition for it rather than the bare definition. Say $x$ is some vector in space and $M$ is some operation on vectors. The definition is: A $n$ × $n$ Hermitian matrix M is called positive-semidefinite if $$x^{*} M x […]

2-norm vs operator norm

I have read that we define the “2-norm” of a matrix as $$\max_i \,{|\sigma_i|},$$ which I have also heard called the “operator norm” (here $\sigma_i$ are the singular values). Also we have the norms $$\|A\| = \left( \sum_{i,j}|a_{ij}|^q \right)^{1/q}$$ for every $q\geq 1$. Do we refer to these as $\|A\|_q$? (For $q=2$, I have heard […]

Eigenvalues of Matrices and Eigenvalue of product of Matrices

If $n\times n$ matrix $A$ has eigenvalues $1,-1$ and $n\times n$ matrix $B$ also has eigenvalues $1,-1$, can I then say something about eigenvalues of $AB$ and $BA$?

Integer matrices with integer inverses

If all entries of an invertible matrix $A$ are rational, then all the entries of $A^{-1}$ are also rational. Now suppose that all entries of an invertible matrix $A$ are integers. Then it’s not necessary that all the entries of $A^{-1}$ are integers. My question is: What are all the invertible integer matrices such that […]