# Questions tagged [matrices]

For any topic related to matrices. This includes: systems of linear equations, eigenvalues and eigenvectors (diagonalization, triangularization), determinant, trace, characteristic polynomial, adjugate and adjoint, transpose, Jordan normal form, matrix algorithms (e.g. LU, Gauss elimination, SVD, QR), invariant factors, quadratic forms, etc. For questions specifically concerning matrix equations, use the (matrix-equations) tag.

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### Prove of disprove.There exists 3x3 real valued matrix such that … [closed]

Prove of disprove this statement. There exists 3x3 real valued matrix B such that $B^2=A$. $$A=\begin{bmatrix} 1 & 0 & 1 \\ 0 & 1 & 1 \\1 & 1& 0 \\ \end{bmatrix}$$
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### Quadratic matrix equation $XAX=B$

Let $A$ and $B$ be two positive semidefinite $n \times n$ matrices. Does the following quadratic matrix equation have a solution in the set of real symmetric matrices? $$XAX=B$$ It's a special case ...
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### Showing that there is unique matrix $B$ such that $B^k=A$ for some $A$

Let $A$ be a $n$ by $n$ real matrix with distinct positive eigenvalues $\lambda_1$,...,$\lambda_n$. And let $k$ be an odd integer. Then, I was able to show that there exists a real matrix $B$ such ...
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### Convexity of $x\mapsto \mathrm{tr}(e^{-E\langle a,x\rangle}bb')$

Let $E$ be a matrix with entries equal to one minus the identity matrix and $a,b$ known vectors. Is the function $$x\mapsto \mathrm{tr}(e^{-E\langle a,x\rangle}bb')$$ convex?
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### Find inverse of specific matrix

So, I have to find an inverse of this matrix: $$\begin{pmatrix} A & B\\ 0 & C \end{pmatrix}$$ where, $A\in M_m(\mathbb{R})$,$B\in M_{mn}(\mathbb{R})$, $C\in M_n(\mathbb{R})$ and $A$ and $C$ ...
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### For $4\times 3$ matrix $M$, for any $3\times4$ matrix $N$, $\exists 0 \neq v \in \mathbb{C}^4$ such that $MNv = 0$.

Define $M := \left(\begin{matrix}1&-1&2\\2&-1&1\\-4&1&0\\3&-2&3\end{matrix}\right)$. Prove or disprove: For every $3 \times 4$ complex matrix $N$, there is a non-zero ...
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### Representing a Bilinear Form in a Matrix

Let $b : \mathbb{R}^n \times \mathbb{R}^n \to \mathbb{R}$ be a bilinear form, and $\langle,\rangle$ be the standard inner product of the Euclidean space and $e_1,...e_n$ be the standard basis for the ...
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### Negative powers of matrix

For any matrix $A$ is it true that, $A^{-n} = (A^{-1})^n = (A^n)^{-1}$? Does this also then apply to powers of diagonalizable matrices? That is, if $A = PDP^{-1}$ then $A^{-n} = PD^{-n}P^{-1}$.
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### Chain rule for differential calculus of 3D rotations

In the paper "A Primer on the Differential Calculus of 3D orientations" by Bloesch et al. they give an example on how to apply the identities derived for the derivatives of various rotation ...
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### Get translational component of particular matrix multiplication

Say I have a $4\times4$ rotation matrix $R$ and $4\times4$ translation matrix $T.$ If I multiply the matrices in this order $T * R$, the translation component of $T$ will not get affected. But if ...
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### Solving a matrix equation using vectorization [closed]

Does anyone know how to solve the following matrix equation for X: $C = X+AXA^T$, where $C$, $X$ are positive semi-definite and $A$ is the adjoint of an SE3 transformation.
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### Show that $g(A)$ is the transformation matrix of the endomorphism $(g(f))$

Let $V$ be a finite-dimensional vectorspace. Given an endomorphism $f:V \to V$. Let $A$ be the transformation matrix of $f$ with respect to a basis $v$ of $V$ and let $g(x)=g_0x^m+ \cdots g_{m-1}x+g_m$...
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### How does matrices magically solve a simultaneous equation?

I know how to solve a simultaneous equation using matrices. But I don't understand how the answer suddenly come out. The stuff I don't understand: 1. What is the logic behind matrix multiplication? ...
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### Theorem about operators on complex vector spaces always having an upper triangular matrix

I'm reading the following proof of a theorem in Linear Algebra done right enter image description here why is it obvious that $(T-\lambda I)u \in U$ from the definition of $U$? Surely if $u \in U$ ...
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### Characterize all real-valued $2\times 2$ matrices with eigenvalues $\pm c$, for $c > 0$.
Characterize all real-valued $2\times 2$ matrices that have as eigenvalues $\lambda_1 = c$ and $\lambda_2 = −c$, for $c > 0$. Use your result to generate a matrix that has its eigenvalues $-1$ and \$...