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 […]

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 $n$?

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 […]

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 […]

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 $X$.

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, \|.\|).$

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 […]

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 […]

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 […]

Suppose that $A$ is a positive definite symmetric matrix (P.D.S.). Now consider the matrix $|A|$, the matrix arrived at by taking the absolute value of all the entries of $A$. Is $|A|$ also P.D.S.? I have been trying to construct a counter example, but I can’t seem to get one. Can someone proved a proof […]

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