# Tagged Questions

For any topic related to matrices. This includes: systems of linear equations, eigenvalues and eigenvectors (diagonalization, triangularization), determinant, trace, characteristic polynomial, adjugate, transpose, Jordan normal form, matrix algorithms (e.g. LU, Gauss elimination, SVD, QR), ...

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### Prove the invertibility of $X^T X$ when $X$ is a (rectangular) Toeplitz-like matrix.

In order to use a minimum squares estimator over some discrete dynamic system parameters, it is necessary to prove that the product $X^T X$ is invertible. Consider the following $N$ by $n+1$ matrix $X$...
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### Are the matrices A and B similar?

$B=\begin{pmatrix}1&7&0\\ \:0&2&7\\ \:0&0&2\end{pmatrix}$ $A=\begin{pmatrix}1&1&5\\ 0&2&0\\ 0&0&2\end{pmatrix}$ They have the same trace,same rank, ...
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### Show that similar matrices have same trace

If $A$ and $B$ are $n\times n$ matrices of a field $F$, then show that $\text{trace}(AB)=\text{trace}(BA)$. Hence show that similar matrices have the same trace. I've done the first part (proving ...
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### Prove that $A$ cannot be invertible if $A^2=0$

Let $A$ be an $n\times n$ matrix for which $A^2=0$. Prove that $A$ can not be invertible. My attempt: Given $A^2 = 0$, this means that $A = 0$. If $A$ is invertible, there must be an $n \times n$ ...
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### Let $A^*$ denote the matrix whose $(ij)$-th entry is $A_{ij}$, $1 ≤ i, j ≤ 5.$

Let $A \in M_5(\Bbb R)$. If $A = (a_{ij})$, let $A_{ij}$ denote the co-factor of the entry $a_{ij}, 1 ≤ i, j ≤ 5.$ Let $A^*$ denote the matrix whose $(ij)$-th entry is $A_{ij}$, $1 ≤ i, j ≤ 5.$ a. ...
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### Flip only one axis of a transformation matrix?

A 4x4 transformation matrix can be multiply to transform a point by translating and rotating it. I have a transformation matrix, however I noticed that the X translation is going to the opposite way ...
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### Finding the expression of the inverse of $(AB)^T$

I know that $(AB)^T$ = $B^TA^T$ and that $(A^T)^{-1}= (A^{-1})^T$ but couldn't reach any convincing answer. Can someone demonstrate the expression.
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### Finding possible determinant values of 3x3 matrix using an equation

Given a 3 x 3 matrix $A$ $4A= A^{7}$ Find the possible values of det(A). I multiplied by $A^{-1}$ both sides and got $4I= A^{6}$ (not helpful) ?? Can you show the right way to solve it ? and how ...
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### Find a,b,c to match the linear transformation matrix?

P.S. Sorry for my bad explanation of the task, it was really hard to translate this into meaningful english For the given linear-transformation $A$ find all possible combinations of a,b,c for which ...
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### Is this matrix decomposition possible?

Given a $2\times2$ matrix $S$ with entries in $\mathbb{Z}$ or $\mathbb{Q}$ , when is it possible to write $S=\frac{1}{3}(ABC+CAB+BCA)$ such that $A+B+C=0$, where $A, B, C$ are matrices over the same ...
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### Pearson Correlation

I have two matrices, which are square but of different size. I want to find correlation between data which is stored in these two matrices. It seems Pearson Correlation Coefficient is applicable for ...