Articles of matrices

Prove or disprove: For $2\times 2$ matrices $A$ and $B$, if $(AB)^2=0$, then $(BA)^2=0$.

The question is in the title. my thoughts on the question: I know that $AB=O$ does not imply that $BA=O$, so my first impression was that it is false. I tried the counter-example I know but it leads to $(BA)^2=O$. (edited) As pointed out by Friedrich Philipp in the comments, $A$ or $B$ is not […]

What can be said if $A^2+B^2+2AB=0$ for some real $2\times2$ matrices $A$ and $B$?

Let $A,B\in M_2(\mathbb{R})$ be such that $A^2+B^2+2AB=0$ and $\det A= \det B$. Our goal is to compute $\det(A^2 – B^2)$. According to the chain of comments on Art of Problem Solving, the following statements are true: $\det(A^2+B^2)+\det(A^2-B^2)=2(\det A^2+\det B^2)$. (Is this well known?) (1) $\implies \det(A^2-B^2)=0$. If $A,B\in M_2(\mathbb{C})$ satisfy $A^2+B^2+2AB=0$, then $AB=BA$. $(A+B)^2=0 \implies […]

Eigenvalues of a $2 \times 2$ block matrix where every block is an identity matrix

I want to consider the following matrix: \begin{bmatrix}\boldsymbol{I}_n & \boldsymbol{I}_n \\\boldsymbol{I}_n & \boldsymbol{I}_n\end{bmatrix} By doing several numerical examples, I recognized that this matrix has $n$ eigenvalues equal to zero and $n$ eigenvalues equal to $2$. Is there any way to prove this for an arbitrary number $n$?

How to rotate a matrix by 45 degrees?

Assume you have a 2D matrix. Ignore the blue squares. The first image represents the initial matrix and the second represents the matrix rotated by 45 degrees. For example, let’s consider a random cell from the first image. a32 (row 3, column 2) Let x be the row and y be the column, so x […]

Proof Strategy – Prove that each eigenvalue of $A^{2}$ is real and is less than or equal to zero – 2011 8C

Remember that we’ve already proven the following, for any real symmetric $n\times n$ matrix $M$: (i) Each eigenvalue of $M$ is real. (ii) Each eigenvector can be chosen to be real. (iii) Eigenvectors with different eigenvalues are orthogonal. (b) Let $A$ be a real antisymmetric $n\times n$ matrix. Prove that each eigenvalue of $A^{2}$ is […]

Does the Schur complement preserve the partial order

Let $\begin{bmatrix} A_{1} &B_1 \\ B_1' &C_1 \end{bmatrix}$, $\begin{bmatrix} A_2 &B_2 \\ B_2' &C_2 \end{bmatrix}$ be symmetric positive definite matrices and be conformably partitioned. If $\begin{bmatrix} A_{1} &B_1 \\ B_1' &C_1 \end{bmatrix}-\begin{bmatrix} A_2 &B_2 \\ B_2' &C_2 \end{bmatrix}$ is positive semidefinite, is it true $(A_1-B_1C^{-1}_1B_1')-(A_2-B_2C^{-1}_2B_2')$ also positive semidefinite? Here $X'$ means the transpose of $X$.

The compactness of the unit sphere in finite dimensional normed vector space

We define $ (\mathbb{R}^m, \|.\|)$ to be a finite dimensional normed vector space with $ \|.\|$ is defined to be any norm in $ \mathbb{R^m}$. Let $S = \lbrace x \in \mathbb{R}^m: \| x\| = 1 \rbrace.$ Prove that $S$ is compact in $ (\mathbb{R}^m, \|.\|).$

Finding conjugacy classes of $PGL_{2}(\mathbb{F}_{q})$

Assume $q$ is odd. How does one go about finding the conjugacy classes of $PGL_{2}(\mathbb{F}_{q})$? I know that for $GL_{2}(\mathbb{F}_{q})$, one can consider the possible Jordan Normal Forms of the matrices and with some luck, choose representatives whose conjugacy class is large enough such that when I sum all the conjugacy class sizes I get […]

Why is the matrix $(I – A)$ theoretically singular?

I’ve the following Matlab code to compute the eigenvector using the inverse iteration (or power) method: A = p * G * D + delta; x = (I − A) \ e; x = x / sum(x); taken from the 4th page of this chapter about the pagerank algorithm by Cleve Moler. First, I don’t […]

Eigenvalues of a tridiagonal stochastic matrix

I’ve tried to calculate the eigenvalues of this tridiagonal (stochastic) matrix of dimension $n \times n$, but I had some problems to find an explicit form. I only know that 1 is the largest eigenvalue. $$M=\dfrac{1}{2}\begin{pmatrix} 1& 1 & & & & \\ 1& 0 &1 & & & \\ & 1 & 0 &1 […]