# Tag Info

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### The determinant of a matrix is "linear in the rows "

The definition of the determinant is a sum of sign weighted products of matrix elements (it seems from the top of the included image). Show that in each of these products exactly one element of row $l$...
1 vote

### How to prove ${\lambda _n}\left( {{X^T}AX} \right) \le {\lambda _n}\left( A \right){\rho ^2}\left( X \right)$?

We may assume that $\lambda_n(X^TAX)>0,$ as otherwise the inequality is obvious. Thus $X$ and $A$ are invertible. We have $$\sigma(X^TAX)=\sigma(XX^TA)=\sigma(A^{1/2}XX^TA^{1/2})$$ where $\sigma(S)$...

### Are differential operators in differential equations additive?

Differentiation is a linear operator from which $$\frac{d}{dx}\left(f(x)+g(x)\right)= \frac{d}{dx}(f(x)) + \frac{d}{dx}(g(x))\tag{1}$$ $$\frac{d}{dx}(c \cdot f(x))= c\cdot \frac{d}{dx}(f(x))\tag{2}$$ ...

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### Given $x\in \mathbb{R}^n$, Is there exist a matrix $A\in \mathbb{R}^{n\times n}$ such that $A$ only have $x$ as the null space?

If $x = 0$, it suffices to take an invertible matrix $A$ (such as the identity matrix). So, I will consider only the case where $x \neq 0$. One convenient way to construct a suitable matrix is to use ...

1 vote

### If we know the direction of the sum of vectors $\frac{\sum_1^\infty w_i}{\|\sum_1^\infty w_i\|}=u$, how can we calculate the expection of the vectos?

To put the question rigorously, assume you have $W$ a random vector. $\left(W_n\right)_{n\ge1}$ are independent observations of $W$. If $\mathbb E\left\|W\right\|^2 < \infty$ then by the law of ...
1 vote
Accepted

### A upper triangular matrix with zeros in the diagonal is nilpotent

Let $V=V_1=F\times F\times F$, $V_2=\{0\}\times F\times F$, and $V_3=\{0\}\times\{0\}\times F$, and $V_4$ to be the zero subspace. Notice that $V_4\subseteq V_3\subseteq V_2\subseteq V_1$. One way to ...
1 vote

### Hyperplane in a complex vector space

Since this has gotten bumped, and may be useful to posterity: Literally, a complex hyperplane in a finite-dimensional complex vector space is (i) a real subspace of real codimension two that (ii) is ...
1 vote

### Determinant of symmetric matrix

It is quite straightforward to arrive at a triangular matrix with the same determinant: After the following elementary row operations: R_1\to R_1-R_2\\ R_2\to R_2-R_3\\ R_3\to R_3-R_4\\ R_4\to R_4-...

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