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Results tagged with linear-algebra
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user 12858
Questions about the properties of vector spaces and linear transformations, including linear systems in general.
16
votes
vector to diagonal matrix
I'm not sure whether it answers your question, but here is a "matrix procedure" to transform the column vector $v$ into a diagonal matrix $D$:
Let $E_i$ be the $n \times n$ matrix with a $1$ on posit …
- 6,614
7
votes
How to prove that a quaternion algebra over ℤₚ is isomorphic to Mat₂(ℤₚ) for p prime?
At the risk of being redundant and repeating earlier answers, let me mention that this is explicitly contained in the book "Elementary number theory, group theory and Ramanujan graphs" by Davidoff, Sa …
- 6,614
5
votes
3
answers
1k
views
adjoint of multiplication operator in a commutative algebra
Dan Popescu asked me the following question, and since I'm not an expert I'm throwing his question on MO.
Suppose that $A$ is a finite-dimensional vector space over an ordered field $k$ with $\operat …
- 6,614
3
votes
1
answer
303
views
ABA-product of matrices and length of chains of principal inner ideals
Let $k$ be a field, $p,q$ positive integers, and let $R$ be the space of $(p \times q)$-matrices over $k$, and $S$ be the space of $(q \times p)$-matrices over $k$. For every matrix $A \in R$, we defi …
- 6,614
3
votes
Accepted
Characterizing the set of self-orthogonal complex vectors
The most natural way to view your set of vectors is in the setting of projective geometry; the vector space $K^n$ (where $K$ is now an arbitrary field, so $K = \mathbb{C}$ for you) can be seen as a pr …
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2
votes
Orthogonal transformations fixing a subspace (setwise)
The full orthogonal group $O(Q)$ is generated by reflections, i.e. involutory isometries fixing a hyperplane pointwise:
$$\pi_v : V \to V : x \mapsto x - \frac{Q(v,x)}{Q(v)} v,$$
where $v$ is an aniso …
- 6,614
2
votes
Accepted
Using permutation matrix to convert a matrix into tridiagonal matrix
The permutation matrix $P$ corresponding to the permutation
$$ \sigma \colon \begin{cases} i \mapsto 2i & \text{ if } i \in [1,n] \\ i \mapsto 2i - 2n - 1 & \text{ if } i \in [n+1,2n] \end{cases} $$
d …
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