Articles of matrices

Proof for $A,B \in M_n(\mathbb{F})$ that if $=tA$ for $0\neq t\in\mathbb{F}$, then $A^n=0$

This question already has an answer here: If A and (AB-BA) commute, show that AB-BA is nilpotent [duplicate] 2 answers

system of equation with 3 unknown

Solve $$\begin{matrix}i \\ ii \\ iii\end{matrix}\left\{\begin{matrix}x-y-az=1\\ -2x+2y-z=2\\ 2x+2y+bz=-2\end{matrix}\right.$$ For which $a$ does the equation have no solution one solution $\infty$ solutions I did one problem like this and got a fantastic solution from @amzoti. Now, I think that if I see another example, I will really get it. EDIT Here is my attempt with rref […]

How to be sure that the $k$th largest singular value is at least 1 of a matrix containing a k-by-k identity

In section 8.4 of the report of ID software, it says that the $k$th largest singular value of a $k \times n$ matrix $P$ is at least 1 if some subset of its columns makes up a $k\times k$ identity. I tried to figure it out but couldn’t be sure of that. Any ideas on […]

How to obtain a possible state space representation of this 2nd order transfer function?

I have this 2nd order transfer function: $$G(s) = \frac{2}{s} + \frac{1}{s+2}$$ And I need to find a possible state space representation in the form of: $$ \frac{dx}{dt} = Ax + bu $$ $$y = c^Tx$$ Matrix A Matrix A is the system matrix, and relates how the current state affects the state change x’ […]

additivity of rank

we know that for all $A,B\in M_n(\mathbb{C})$ : $$ rank (A+B)\leq rank(A)+rank (B) $$ see here for a simple proof, but for which condition on the coefficients of $A$ and $B$ we can obtain a perfect equality. more simply if we assume that $rank(A)=1$ what will be the condition on $B$ to have : $$ […]

How do I write this matrix in Jordan-Normal Form

I have the matrix $A=\begin{pmatrix}2&2&1\\-1&0&1\\4&1&-1\end{pmatrix}$, I want to write it in Jordan-Normal Form. I have $x_1=3,x_2=x_3=-1$ and calculated eigenvectors $v_1=\begin{pmatrix}1\\0\\1\end{pmatrix},v_2=\begin{pmatrix}1\\-4\\5\end{pmatrix},v_3=\begin{pmatrix}0\\0\\0\end{pmatrix}$. But, the matrix $Z=\begin{pmatrix}1&1&0\\0&-4&0\\1&5&0\end{pmatrix}$ is not invertible since $\text{det}(Z)=0$. Does this mean the matrix cannot be written in JNF or do I need to find different eigenvectors? I have tried to find different eigenvectors, but […]

Is there meaning for $\bf uv^T$?

Let $\bf{u,v}$ be two column vector in $\mathbb R^n$, which can be represented by $n\times1$ matrix. $\bf u^T v$ is the inner product of $\bf u,v$, then is there meaning for $\bf uv^T$, which is a $n\times n$ matrix? Thanks.

Cramer's Rule for Inverse

I am trying to find the inverse of the following matrix using Cramer’s rule:$\begin{bmatrix}2&0&0&0\\ 1&-2&1&-1\\ 0&2&2&3 \\1&0&1&1 \end{bmatrix}$. If this was a $3\times3$, I would know the process but I am completely stuck since this is a $4 \times 4$. My question is, how do you even begin to calculate $C_{1,1}, …., C_{m,n}$? I tried […]

Can we specify all row equivalent matrices of a given matrix?

Say we have a RREF matrix like $$\begin{bmatrix} 1 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0\end{bmatrix}$$ From this matrix, is there some way of specifying ALL of the matrices which are row equivalent to this one?

Find the inverse of a matrix with a variable

$$X= \begin{pmatrix} 2-n & 1 & 1 & 1 & \ldots & 1 & 1 \\ 1 & 2-n & 1 & 1 & \ldots & 1 & 1 \\ \vdots & \vdots & \vdots & \vdots & \ddots & \vdots & \vdots \\ 1 & 1 & 1 & 1 & \ldots & 2-n […]