Articles of matrices

why does the reduced row echelon form have the same null space as the original matrix?

What is the proof for this and the intuitive explanation for why the reduced row echelon form have the same null space as the original matrix?

the representation of a free group

A group $G$ is generated by $\begin{pmatrix}1&n\\0&1\end{pmatrix}$ and $\begin{pmatrix}1&0\\n&1\end{pmatrix}$, then we know $G\cong \mathbb{F}_2$ which is a free group generated by two elements. Now I consider the representation: $G\to GL(2,\mathbb{R})$, it is necessary that the image of $\begin{pmatrix}1&n\\0&1\end{pmatrix}$ is a triangular matrix under conjugation? Thanks in advance.

Orthogonal Decomposition

[Ciarlet 1.2-2] Let $O$ be an orthogonal matrix. Show that there exists an orthogonal matrix $Q$ such that $$Q^{-1}OQ\ =\ \left(\begin{array}{rrrrrrrrrrr} 1 & & & & & & & & & & \\ & \ddots & & & & & & & & & \\ & & 1 & & & & & & & […]

Behavior of the spectral radius of a convergent matrix when some of the elements of the matrix change sign

I want to prove (or disprove) the following statement: If $A$ is a square matrix with non-negative elements that has spectral radius less then $1$, then any matrix obtained from $A$ by arbitrarily changing the sign of the elements has the same property. This problem appeared recently when studying the convergence of some matrices and […]

Connected components of a given subspace of $M_{n \times n}(\mathbb{R})$.

This question is motivated by this question, which gave me quite a headache today. Context: I posted originally what I thought was a quick proof using the derivative of the given function. It was intended to be a straightforward answer using a well-known strategy of proving something is constant and equal to some other thing […]

Why does the Gaussian-Jordan elimination works when finding the inverse matrix?

In order to find the inverse matrix $A^{-1}$, one can apply Gaussian-Jordan elimination to the augmented matrix $$(A \mid I)$$ to obtain $$(I \mid C),$$ where $C$ is indeed $A^{-1}$. However, I fail to see why this actually works, and reading this answer didn’t really clear things up for me.

Eigenvalues of adjugate matrix of a singular matrix

Given a singular matrix $A$, find the eigenvalues of the adjugate matrix of $A$. The same question with $A$ being invertible is trivial since $A\operatorname{adj}A=(\operatorname{adj}A)A=(\det A) I$. If $\operatorname{rank}A\leq n-2$, it is well-known that $\operatorname{adj}A=0$ and $0$ is the only eigenvalue. It remains to deal with the case $\operatorname{rank}A= n-1$. It is easy to check […]

When is the geometric multiplicity of an eigenvalue smaller than its algebraic multiplicity?

I was kinda crushed to discover that two different matrices with different properties can actually share the same characteristic polynomial ($-\lambda^3-3\lambda^2+4$): $A=\begin{pmatrix} 1 & 2& 2\\ -3 &-5 &-3 \\ 3& 3 & 1 \end{pmatrix} , B=\begin{pmatrix} 2 & 4& 3\\ -4 &-6 &-3 \\ 3& 3 & 1 \end{pmatrix}$ $A$ has an eigenline and […]

How do collinear points on a matrix affect its rank?

Consider the matrix \begin{matrix} x_1 & y_1 & 1 \\ x_2 & y_2 & 1 \\ x_3 & y_3 & 1 \\ \end{matrix} what effect does $({x_1},{y_1})$,$({x_2},{y_2})$,$({x_3},{y_3})$ being collinear have on the rank of the above matrix ?

What is the intuitive meaning of the adjugate matrix?

The definition of the adjugate matrix is easy to understand, but I have never seen it used for anything. What is the intuitive meaning of this matrix? Are there examples of applications which may shed light on its conceptual meaning? I would be especially interested to hear examples of usage in representation theory or other […]