# Tagged Questions

Linear algebra is concerned with vector spaces of all dimensions and linear transformations between them, including systems of linear equations, bases, dimensions, subspaces, matrices, determinants, traces, eigenvalues and eigenvectors, diagonalization, Jordan forms, etc. For questions specifically ...

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### Power of a matrix, given its jordan form

Can someone please explain how to find the power of a matrix $A$, given $A=MJM^{-1}$ where the matrix $J$ is in the Jordan canonical form? Or else please explain how to find the powers of a matrix ...
2answers
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### Proving that $A$ is diagonalizable

Let $A\in\mathbb{C}^{n\times n}$ be a $n$ by $n$ matrix such that $A^k = I$ for some natural number $k$. Find a nonzero annihilating polynomial of A and prove that A is diagonalizable. I will say ...
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### Showing unitary similarity of these two matrices

Let $A \in B(H)$ for a Hilbert space $H$, and $\alpha \in \sigma_{p}(A)$, the point spectrum of $A$. Suppose ker$(\alpha I-A)$ is not a reducing subspace of $A$ then $A$ has the following matrix ...
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### Finding eigenvalues and “eigenmatrices”.

On the space of $2\times 2$ matrices, let $T$ be the transformation that transposes every matrix. Find the eigenvalues and "eigenmatrices" for $A^T =\lambda A$. By taking determinants on the left and ...
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### Some simple matrix identities

I've recently been learning some linear algebra and I've isolated what seem to be some important matrix relations (often used tacitly). I would be most grateful if someone could just check that I have ...
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### Question about a definition

There was a definition on my notebook. But sadly I cant read (...) part. What do we call $w_1,w_2,w_3...w_k$? Let V be a vector space on field F and $w_1,w_2, w_3..$ are subspaces of V. for any ...
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### Can I turn $Ax=b$ into $Ax=0$?

For a system of equations $$\begin{bmatrix}d_1 & d_2 & \dots & d_n \end{bmatrix} \begin{bmatrix}u_1\\u_2\\ \vdots \\ u_n \end{bmatrix} = d_{n+1}$$ where each $d$ is a column of ...
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### Spanning Set Of $V$ After Linear Transformation Will Span $U$

let there be a Linear Transformation $T:V \rightarrow U$ and $B={v_1,...,v_n}$ a spanning set of $V$, so $C=T(v_1),...,T(v_n)$ will span $U$. Is it right because: 1. there is only one linear ...
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### Minimizing an error function by deriving a system of linear equations

Consider the following formula: $$E(\mathbf{w}) = \frac{1}{2}\sum_{n=1}^{N}\{y(x_n,\mathbf{w})-t_n\}^2$$ where $\mathbf{w}$ is a vector of weights; $x_n$ and $t_n$ come from two vectors of length ...
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### prove that $\sum_{k=1}^\infty|x_k y_k|$ converges

Let $V$ be the space of real sequences $x_k$ so that $\sum_{k=1}^\infty x_k^2$ converges. Let $\langle x,y\rangle=\sum_{k=1}^\infty x_k y_k$ Prove that $\sum_{k=1}^\infty |x_k y_k|$ converges My ...
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### Construct and apply a rotation matrix by doing the following

Create a 2x2 rotation matrix $A \ne I$. Determine, showing all work, the location of point $(3, 2)$ when it is rotated using the linear transformation generated by the matrix. Also, demonstrate, ...
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### If $B^T$ consists of a basis of $\mathrm{im} (A)^\perp$, then $\mathrm{im}(A)=\ker (B)$?

Well basically, the question is in the title: Suppose we have $A\in\mathbb{R}^{n\times m}$ with rank $d$ and we fix a basis $(b_1,\ldots,b_{n-d})$ of $\mathrm{im}(A)^\perp$. Let ...
2answers
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### Subspace of matrices AB = BA

I'm stuck with the following exercise: Let A be an $n\times n$ diagonal matrix with characteristic polynomial: $$\prod_{i=1}^{k}(x-c_{i})^{d_{i}}$$ where $c_{1},...,c_{k}$ are distinct. Let $W$ ...
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### How to mathematically determine if the magnitude of a cross product is up/down(positive/negative?)?

So, I'm a newbie at complex vector math. I'm working on a 2D physics engine, and my issue is, with angular acceleration from torque, is it supposed to be positive or negative? I understand the right ...
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### Solutions of $XA=XAX$.

All matrices are real and $n \times n$. The matrix $A$ is given. I am interested in solving $XA=XAX$. In particular, I would like some characterization of matrices that satisfy this equation. For ...
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### Linear Transformation On Basis

What a Linear Transformation does on a basis? if the Linear Transformation is 1-1 and onto so every element of the basis goes to element of the basis of the other vector space? and what if it is not ...
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### Proving result on spectral radius

How do I prove that $$\rho(A)=\inf\limits_{\text{operator norms}}\|A\|,$$ $\rho$ being the spectral radius, $A$ being a complex $n\times n$ matrix and operator norms being induced from vector norms by ...
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314 views

### How to find eigenvalues of the following block circulant matrix

I have a block matrix of size PN x PN of the form: Where A and C are P x P matrices. I would like to find the eigenvalues of the matrix B, that is where
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47 views