# Tagged Questions

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### Prove that the direct sum of a symmetric and skew symmetric matrix belongs to $M_n(K)$ using $A_{ij}$ and $A_{ji}$ notation.

Basically Let $M_n(K)$ be an $n\times n$ matrix of a $K$ vector space. $U =\{A\in M_n(K)\;|\;A_{ij}=A_{ji}\}$ $W =\{A\in M_n(K)\;|\;A_{ij}=−A_{ji}\}$ So I don't understand my mark scheme. It says ...
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### Show that 2 matrices belong to a square matrix by taking the transpose. Vector spaces

Let $M_n(K)$ be an $n\times n$ matrix of a K vector space. \begin{align} U &= \{A ∈ M_n(K) | A_{ij} = A_{ji} \} \\ W &= \{A ∈ M_n(K) | A_{ij} = -A_{ji} \} \end{align} Prove that $U$ and $W$ ...
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### Projective special linear groups

Is it known if $PSL(2,\, F)$, with $F$ a field of prime $p$ characteristic (maybe with all proper subfields of finite order), is co-hopfian? I've searched everywhere but found nothing. Definition A ...
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### Is there a way to determine the matrix of $\Lambda^k(T)$ given the matrix of $T$?

Let $T$ be an endomorphism of a finite dimensional vector space $V$. Suppose that $(v_1,\ldots v_n)$ is an ordered basis of $V$. And let $[T]$ be the matrix of $T$ with respect to this basis. Is ...
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### A problem on matrices over a commutative ring

Let $M_{m,n}(R)$ denote an $m\times n$ matrix with each entry over a commutative ring $R$, $m\leq 2\leq n$, and there is a matrix $\mathbf{B} = M_{m,n}(R)$. $\mathbf{B}\mathbf{s} = \mathbf{a}$, where ...
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### Is there some fast and efficient way for solving $x$

Let $b$ be a given constant scalar between 0 and 1, and $A$ a given $N \times N$ transitional probability matrix (i.e., each row sum of $A$ is 1, and $0\le {A}_{(i,j)} \le 1$). Let $A\circ A$ denote ...
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### Matrix restoring (modulo n)

Let $m,n\geqslant 2$, and $A\in \mathcal{M}_n(\mathbb Z)$ such that $\det A \equiv 1 \pmod m$. Does it (necessarily) exist $M\in \mathrm{GL}_n(\mathbb Z)$ such that $A\equiv M \pmod m$?
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### If $A\ne 0$ is a square matrix over a commutative ring with $\det A=0$, then its null space contains an element whose components are minors of $A$

Let $R$ denote a commutative ring and $A\ne 0$ a $n\times n$ matrix over $R$ with $\det A=0$. Then there exists a $x\in\ker A\setminus\left\{0\right\}$ such that all components of $x$ are minors of ...
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### Show that $\lambda_1 = \min \{ Q(u) \mid \|u\| = 1 \}$ and $\lambda_m = \max \{ Q(u) \mid \|u\| = 1 \}$

Let $V$ a vector space over $K$ and $Q(u) = \langle u, Tu \rangle$ a quadratic form. $T$ is a symmetric operator. The eigenvalues of $T$ are sorted by size $\lambda_1 < \dots < \lambda_m$. How ...
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### Write Matrix $A$ to $A = \sum_{i=1}^{3} \lambda_i P_i$

Let $A$ be a Matrix: $$A = \begin{pmatrix} 1 & 0 & 3i \\ 0 & -3 & 0 \\ -3i & 0 & 1 \end{pmatrix}$$ Now I want to write $A$ as $$A = \sum_{i=1}^{3} \lambda_i P_i$$ I ...
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### how can I obtain the lower bound for this case?

For an arbitrary pair of orthogonal bases $\phi$, $\varphi$ that construct the matrix A and which of them are $n \times n$. mutual coherence for matrix $A$ is defined as te maximal inner product ...
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### Application of Schur's Lemma

Given three matrices , $B_1 \in Mat_{n_1}(k)$, $B_2 \in Mat_{n_2}(k)$, and $C \in Mat_{n_1 \times n_2}(k)$, where $k$ is a perfect field, and $B_1$ and $B_2$ are irreducible matrices. Required to ...
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### How to find a basis for a tricky 2x 2 matrices vector space

Consider the vector space of 2 x 2 matrices :\begin{bmatrix}a&b\\0&c\end{bmatrix} such that a and c are rational numbers and b is a real number with rational numbers as the field of this ...
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### How to prove that the inverse of a matrix is unique?

The ring of matrix is not an integral domain. How to prove that the inverse is unique?
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### Is the operation associative

Is it known that the multiplication of matrices is a associative operation ? So,is the relation $(A \cdot B) \cdot C=A \cdot (B \cdot C)$ true?? ($A,B,C$ are matrices)
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### Maximal Ideals of Matrix Ring

Let $D$ be a division algebra. I'm trying to show that all simple $M_n(D)$ modules are isomorphic to $D^n$. I know that $M_n(D)=D^n\oplus D_n \oplus \dots \oplus D^n$ (n terms). I have that if $S$ is ...
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### suggest me a example for non singular, conjugate-symmetric sesquilinear form ????

I only know that fact that the matrix corresponding to the non singular, conjugate-symmetric sesquilinear form is a unitary matrix. and SU_n(q) is the unitary groups is the collection of the ...
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### center of invertible matrices

find the center of the group of invertible 2 x 2 matrices with real entries. Attempt: By definition, the center of a group Z(G), is where all the elements are commutative. If G = { invertible 2 x 2 ...
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### Question about $2 \times 2$, determinant one matrices over a Euclidean domain

Let $R$ be a Euclidean domain. Let $a,b,c, d \in R$ with $ad - bc = 1_R = 1$. That is, the matrix:  A = \left( \begin{array}{cc} a & b \\ c & d \\ \end{array} \right) \in ...