Articles of matrix rank

Rank of the differential

Let $f:\mathbb R^n \to \mathbb R^n$ such that $f$ maps roots of a polynomial to its coefficients. Meaning: if $(x-x_1)(x-x_2)…(x-x_n)=x^n+a_1x^{n-1}+a_2x^{n-2}+…+a_n$ then $f\begin{pmatrix} x_1 \\x_2\\x_3\\ \vdots \\x_n \end{pmatrix} =\begin{pmatrix} a_1\\a_2\\a_3 \\ \vdots \\a_n \end{pmatrix}$ Show that the rank of the differential of $f$ is equal to the number of different roots. $rank(Df)=cardinal\{x_1,…x_n\}$

Find a basis for the null space of a linear transformation T when we are given the spanning set of the range of T

Let $T:\mathbb{R}^{5}\rightarrow\mathbb{R}^{4}$ be a linear transformation of the form $Tx=Ax$ where $A$ is a matrix of appropriate size with entries in the field of real numbers. Suppose that the range of T is spanned by the vectors $\alpha_{1}=\left(1,1,2,4\right),\alpha_{2}=\left(2,-1,-5,2\right),\alpha_{3}=\left(1,-1,-4,0\right),\alpha_{4}=\left(2,1,1,6\right)$ . Find a basis for the null space of $T$

Rank of square matrix $A$ with $a_{ij}=\lambda_j^{p_i}$, where $p_i$ is an increasing sequence

Let $$ A = \begin{bmatrix} \lambda_1^{p_1} & \lambda_2^{p_1} & \cdots & \lambda_n^{p_1} \\ \lambda_1^{p_2} & \lambda_2^{p_2} & \cdots & \lambda_n^{p_2} \\ \lambda_1^{p_3} & \lambda_2^{p_3} & \cdots & \lambda_n^{p_3} \\ \vdots & \vdots & \ddots & \vdots \\ \lambda_1^{p_n} & \lambda_2^{p_n} & \cdots & \lambda_n^{p_n} \end{bmatrix}$$ where $p_1=0$ and $p_k > p_i$ for $i < k […]

rank of block triangular matrix and its relation to the rank of its diagonal blocks

Prove that $$rank\begin{pmatrix}A & X \\ 0 & B \\ \end{pmatrix}\ge rank(A) + rank(B)$$ where $$A,B\in \mathbb C^{m \times m}$$. I know the intuition behind it (i.e. maximal independent rows, etc.), but I am looking for a formal proof. I have tried QR decomposition of A and B, then broke the block triangular matrix up […]

If $A$ is an $m\times n$ matrix and $B$ is an $n\times m$ matrix such that $AB=I$, prove that rank$(B)=m$

If $A$ is an $m\times n$ matrix and $B$ is an $n\times m$ matrix such that $AB=I$, prove that rank$(B)=m$. I am not sure where to begin with this proof. I have that rank$(AB) = m$, but I can’t find anything to help me get further. Can anyone help me out? Thanks.

Small perturbation of linear transformation cannot decrease its rank

Suppose a linear transformation $T:\mathbb{R}^n\rightarrow \mathbb{R}^m$ has rank $k$.Show that there exists a $\delta>0$ such that if for a linear transformation $S:\mathbb{R}^n\rightarrow \mathbb{R}^m$ we have that $\left \|S-T \right \|<\delta$, then $rank(S)\geqslant k$. My hunch tells me that the proof should be simple, but a cannot figure this out. I’d appreciate a small hint.

Relationship: Rank of a matrix $\leftrightarrow$ # of eigenvalues

Can someone tell, why the number of the nonzero eigenvalues (counted according to their algebraic multiplicities) of a matrix of type $A^{*}A$, where $A$ is an arbitrary real or complexvalued matrix, is equal to the rank of $A$ ? Here, in step 2, it seems to me that exactly this assertion is made, and I […]

For nonzeros $A,B,C\in M_n(\mathbb{R})$, $ABC=0$. Show $\operatorname{rank}(A)+\operatorname{rank}(B)+\operatorname{rank}(C)\le 2n$

Let $A,B,C\in M_n(\mathbb{R})$ be nonzero matrices such that $ABC=0$. How can we prove that $\operatorname{rank}(A)+\operatorname{rank}(B)+\operatorname{rank}(C)\le 2 n$ ? I can prove this for two matrices, but in this case, i can’t!

For a set of symmetric matrices $A_i$ of order p, show that if the sum of their ranks is p, $A_iA_j=0$

Here’s what I know. Matrices $A_i$ for $i=1,…,k$ are all symmetric p by p matrices. $\sum\limits_{i=1}^k A_i = I_p$ where $I_p$ is the p by p identity matrix $\sum\limits_{i=1}^k rank(A_i) = p$ With this, I have to find a way to show that for all $i \neq j$, $A_iA_j=0$. I assume this is solved by […]

Rank of transformation $Y=AX-XA$

Consider vector space $V$ consist all $n \times n$ matrix (real or complex). What is the rank of the linear transformation $f(X)=AX-XA$ ($A\in V$)? ($A$ is a given matrix, which means we can have information about it) I have tried to consider the basis of $V$ but it doesn’t work. EDITED: $\operatorname{rank} f = n^2 […]