# 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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### What is the most rigorous definition of a matrix?

Is matrix just a rectangular array of symbols and expressions, or one can define it in a more formal way?
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### A symmetric matrix whose square is zero

I was once asked in an oral exam whether there can be a symmetric non zero matrix whose square is zero. After some thought I replied that there couldn't be because the minimal polynomial of such a ...
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### Sum of all elements in a matrix

The trace is the sum of the elements on the diagonal of a matrix. Is there a similar operation for the sum of all the elements in a matrix?
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### What's the name for the property of a function $f$ that means $f(f(x))=x$?

I can think of several examples of functions such that twice application of the function is equivalent to no application of it. Additive inverse Multiplicative inverse Fourier transform Complex ...
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### Traces of all positive powers of a matrix are zero implies it is nilpotent

Let $A$ be an $n\times n$ complex nilpotent matrix. Then we know that because all eigenvalues of $A$ must be $0$, it follows that $\text{tr}(A^n)=0$ for all positive integers $n$. What I would like ...
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### Expected Value of a Determinant

Suppose that I construct an $n \times n$ matrix $A$ such that each entry of $A$ is a random integer in the range $[1, \, n]$. I'd like to calculate the expected value of $\det(A)$. My conjecture is ...
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### Why do the $n \times n$ non-singular matrices form an “open” set?

Why is the set of $n\times n$ real, non-singular matrices an  open subset of the set of all $n\times n$ real matrices? I don't quite understand what "open" means in this context. Thank you.
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### How to check if a symmetric $4\times4$ matrix is positive semi-definite?

How does one check whether symmetric $4\times4$ matrix is positive semi-definite? What if this matrix has also rank deficiency: is it rank 3?
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### Does this expression have a (“better”) determinant form?

[Edit I've found a determinant that satisfies the letter of the previous version of this question, but not its "Cayley-Menger" spirit.] A tetrahedron with face areas $w$, $x$, $y$, $z$ and "pseudo-...
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### Finding the Exponential of a Matrix that is not Diagonalizable

Consider the $3 \times 3$ matrix $$A = \begin{pmatrix} 1 & 1 & 2 \\ 0 & 1 & -4 \\ 0 & 0 & 1 \end{pmatrix}.$$ I am trying to find $e^{At}$. The only tool I have to find ...
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### Variety of Nilpotent Matrices

Let $k$ be an algebraically closed field and view $M_n(k)$ as $\mathbb{A}^{n^2}$. $A\in M_n(k)$ is nilpotent if and only if $A^n=0$. Since the equation $A^n=0$ is given by $n^2$ polynomial equations,...