# 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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### Why is $<T\vec x,\vec y>=<\vec x,T^*\vec y>$ for hermitian matrix T?

In this video in 5:15 there's a proof that every hermitian matrix has real eigenvalues. I don't understand the step: $<\vec x, L\vec y>=<L^*\vec x,\vec y>$. I know that I can pull out ...
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### Matrix derivative (chain rule application)

Let $x$, $y$ by vectors s.t. $x=f(y)$ and let $B$ be a constant matrix. What is $\frac{\partial x'Bx}{\partial y}$? The partial derivative $\frac{\partial x'Bx}{\partial x}=2Bx$ and we need to use ...
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### What is meant by $\mathbb R^m$ in this context?

I'm studying some introductory Linear Algebra text. So far, it's explained $\mathbb R^2$, $\mathbb R^3$, $\mathbb R^n$ — all understood. Then $\mathbb R^m$ came out of the blue with no background ...
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### Let $\det (A + \alpha I) = 0$ for all $\alpha \in \mathbb{C}\backslash \left\{ 0 \right\}$. Can we say that $\det (A) \ne 0$?

Let $A \in {M_n}$ and $\det (A + \alpha I) = 0$ for all $\alpha \in \mathbb{C}\backslash \left\{ 0 \right\}$. Can we say that $\det (A) \ne 0$?
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### Problem calculating the sine of a matrix

Given the matrix $A=\begin{pmatrix}-\frac{3\pi}{4} & \frac{\pi}{2}\\\frac{\pi}{2}&0\end{pmatrix}$, I want to calculate the sine $\sin(A)$. I do so by diagonalizing A and plugging it in the ...
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### Multiple linear combinations

Lets say I have a vector $v$ and $n$ vectors $u_1,\; ... \;, u_n$ Is there a fast way to know if there are more than 1 or just 1 possibility to express $v$ as combination of $u$'s ? The information ...
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### Existence of solution to underdetermined linear system with variable coefficient matrix.

I'm trying to think through a network flow problem, and while I could probably shuffle this into a form that a linear programming method would work, it feels like there ought to be something more ...
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### Find vectors that span the kernel of $\begin{bmatrix}1&2\\3&4\end{bmatrix}$

I have the following matrix: \begin{bmatrix}1&2\\3&4\end{bmatrix} and I'd like to find the vectors that span the kernel. The book I'm reading isn't helping me understand this concept at ...
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### Eigenvectors of an approximated symmetric matrix

A $3 \times 3$ symmetric matrix has the form $$S=\begin{bmatrix} x & y & z \\ y & w & 0 \\ z & 0 & u \end{bmatrix}$$ While finding eigenvalues, I had to approximate the ...
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### Definite Integral of Kronecker product of matrices

How to prove: $$\int_k^{k+1}\int_k^{k+1} (A⊗B) dxdy = \int_k^{k+1}\int_k^{k+1} A dx⊗\int_k^{k+1}\int_k^{k+1} Bdy$$, where k is an integer and A and B are matrices consisting of variables x and ...
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### Finding the inverse matrix

I have these matrices: Find the inverse matrices: \begin{bmatrix} 1 & 1 & 0& 0&\dots & 0& 0\\0 & 1 & 1& 0&\dots & 0& 0 \\0 & 0 & 1& 1&...
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### Relation between row-space and column-space vectors

Let $A$ be any $n$ by $m$ matrix. $V$ is an orthonormal vector in column-space of $A$. $U$ is an orthonormal vector in row-space of $A$. Now, why is the following relation True? $$AV=U\Sigma$$ , ...