Articles of matrices

Maximize trace of matrix equation given two constraints

Let $\mathbf{Q}$ be a rotation matrix and $\mathbf{A}$ and $\mathbf{B}$ be two real-valued matrices of the same size. I want to maximize the function $$ f(\mathbf{Q})=tr\;\mathbf{QA} \qquad \text{s.t.} \; \mathbf{Q}’\mathbf{Q} = \mathbf{I} \quad \text{and} \quad tr\; \mathbf{QB} \geq 0 $$ If only the orthogonality constraint is imposed, the solution is $$ \mathbf{Q} = \mathbf{VU}’ $$ […]

Possible eigenvalues of a matrix $AB$

Let matrices $A$, $B\in{M_2}(\mathbb{R})$, such that $A^2=B^2=I$, where $I$ is identity matrix. Why can be numbers $3+2\sqrt2$ and $3-2\sqrt2$ eigenvalues for the Matrix $AB$? Can be numbers $2,1/2$ the eigenvalues of matrix $AB$?

If two real symmetric square matrices commute then does they have a common eigenvector ?

Let $A,B$ be real symmetric $n \times n$ matrices such that $AB=BA$ , then is it true that $A,B$ have a common eigenvector in $\mathbb R^n$ ?

Explicit formula for inverse matrix elements

Let $A$ be an $n \times n$ invertible matrix with \begin{align} \left(\begin{array}{ccc} a_{11} & \cdots & a_{1n} \\ \vdots & \ddots & \vdots \\ a_{n1} & \cdots & a_{nn} \end{array}\right)^{-1}= \left(\begin{array}{ccc} b_{11} & \cdots & b_{1n} \\ \vdots & \ddots & \vdots \\ b_{n1} & \cdots & b_{nn} \end{array}\right) \end{align} Is there an explicit formula […]

to prove $f(P^{-1}AP)=P^{-1}f(A)P$ for an $n\times{n}$ square matrix?

let $f(X)$ be a polynomial and let $A$ be $n\times n$ matrix.We have to show that for any $n\times n$ invertible matrix $P$, $f(P^{-1}AP)=P^{-1}f(A)P$ and that there exist a unitary matrix $U$ such that both $U^*AU$ and $U^*f(A)U$ are upper triangular, where $U^*$ is conjugate transpose of $U$ and $P^{-1}$ is inverse of $P$ ..(m […]

DE solution's uniqueness and convexity

I am lost and don’t know how to prove the following: If $M$ is a positive definite symmetric square matrix and if $\overrightarrow {v}(t)$ is a solution of: $$\overrightarrow {v’}(t) = M\overrightarrow {v}(t),\qquad t\in[0,T]$$ Then, 1) $\phi(t) = \ln(\|\overrightarrow {v(t)}\|^2) $ is a convex funciton, 2) Solution of the differential equation is unique.

Why is the reduced echelon form of a set of independent vectors, the identity matrix?

If a matrix has linearly independent rows, then its reduced echelon form is the identity matrix. I haven’t found a concise explanation for this… I have the whole notion in my head but I cannot express this in words. Can someone explain it?

How to do $\frac{ \partial { \mathrm{tr}(XX^TXX^T)}}{\partial X}$

How to do the derivative \begin{equation} \frac{ \partial {\mathrm{tr}(XX^TXX^T)}}{\partial X}\quad ? \end{equation} I have no idea where to start.

Finding Smith normal form of matrix over $ \mathbb{R} $

I am trying to find the Smith normal form of matrix over $ \mathbb{R} [ X ]$ of the 4×4 matrix $$M =\begin{pmatrix} 2X-1 & X & X-1 & 1\\ X & 0 & 1 & 0 \\ 0 & 1 & X & X\\ 1 & X^2 & 0 & 2X-2 \end{pmatrix}$$ I know […]

Number of positive, negative eigenvalues and the number of sign changes in the determinants of the upper left submatrices of a symmetric matrix.

How do we prove that the number of sign changes in the sequence of the determinants of the upper-left matrices of a symmetric matrix $A$ corresponds to the number of positive and negative eigenvalues of $A$? Progress I know that it is true for symmetric matrices that if all the upper left determinants are positive, […]