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How to perform multiple rotations by direction vectors?
[I'm not sure whether I make the right choice of wording.] I am looking for some performant algorithm for a rotation of multiple points in $\mathbb{R}^2$ (and maybe also $\mathbb{R}^3$ with one ... matrices complex-numbers vectors rotations quaternions- 111
Matrix norm (submultiplicative constant)
Is it true that we can always find some $k>0$ such that $||XY||\leq k||X||\cdot||Y||$ holds? $||\cdot|| $ is any matrix norm. I haven't learn functional analysis yet. real-analysis linear-algebra matrices numerical-linear-algebra matrix-analysis- 11
Eigenvalues of block matrix $\begin{pmatrix} A & B \\ B &-A \end{pmatrix}$
It is known that the set of eigenvalues of the following block matrix $$ C = \begin{pmatrix} A & B \\ B & A \end{pmatrix} $$ is the union of the eigenvalues of the matrices $A + B$ and $A - B$.... matrices eigenvalues-eigenvectors block-matrices- 337
Using Euler Angles to change grain orientations
I am looking to manually orientate the direction of grains to an angle that I choose. I have grains that are characterised by Euler angles using Bunge notation. Let me take an arbitrary orientated ... matrices rotations- 11
Boundednesss for norm of a matrix
Let $A$ be a $n\times n$ matrix with some $n\in\mathbb{N}$. Then $\left\|A\right\|\le L$ if and only if $$-LI_n\preceq A\preceq LI_n,$$ where $X\preceq Y$ means that $Y-X$ is a positive semidefinite ... matrices normed-spaces- 101
is the mix of convex and linear functions always convex function?
I want to prove that the following composed function $g \circ L$ is always (strictly) convex : \begin{alignat*}{3} &g&&(t&&) && =-\log(1-e^{-t}) \qquad && \text{... matrices convex-analysis convex-optimization log-likelihood convexity-inequality- 21
Cholesky decomposition question
I am studying for a exam and I thought about practicing the Cholesky decomposition. If a matrix $A = A^{T}$ , the main diagonal of $A$ has only positive elements and in every row the absolute value of ... linear-algebra matrices matrix-decomposition cholesky-decomposition- 113
What does $\mbox{diag}(A)$ denote?
Let $A$ be a $2 \times 2$ matrix. What does $\mbox{diag}(A)$ denote? It can't refer to a block-diagonal matrix, so does it basically mean $A$ with anything but the diagonal set to $0$? linear-algebra matrices notation- 113
Sum of squared symmetric matrices is the identity matrix
Let $A_1, A_2,\ldots,A_n$ be $m\times m$ real symmetric matrices such that \begin{equation} A_1^2 + A_2^2 + \dots + A_n^2 = \alpha I_m. \end{equation} Can we prove that $A_i^2 = \alpha_i B_i$, where $... matrices symmetric-matrices- 1
Complexity of SVD computation assuming knowing left singular vectors and singular values
Suppose we know the left singular vectors $\mathbf{U}$ and singular values $\mathbf{\Sigma}$ for the matrix $\mathbf{A}$ with SVD decomposition $\mathbf{U}\mathbf{\Sigma}\mathbf{V}^{\mathrm{H}}$, then ... linear-algebra matrices optimization computational-complexity svd- 69
Can we generate a block skew-symmetric matrix by a operator
Suppose we have following speical block matrix $$ X = \begin{bmatrix} 0 & X_3 & -X_2\\ -X_3& 0& X_1\\ X_2 &-X_1 &0 \end{bmatrix} $$ where $X1,X2,X3 \in R^{n \times n}$ and not ... linear-algebra matrices operator-algebras block-matrices skew-symmetric-matrices- 21
Is this definition for a strictly triangular system from my textbook incorrect?
Definition from my textbook (Linear Algebra with Applications by Steven J. Leon, 9edg): A system is said to be in strict triangular form if , in the $k$th equation, the coefficients of the first $k-1$... linear-algebra matrices definition- 315
Matrix equation with constraints
I have to estimate the square $n\times n$ matrix $A$ and I know that it must solve the equation: $$(BA)^* + (BA) = \Omega$$ where $B$ is also a $n\times n$ matrix and where $\Omega$ is a Hermitian $n\... matrices matrix-equations matrix-decomposition symmetric-matrices- 669
Upper bound on $\|A^{-1}\|_{\infty}$ for $A$ with unit length columns.
Consider a non-singular $A \in \mathbb{R}^{n\times n}$ with $\|\mathrm{col}_i(A)\| = 1$ ($i \in \{1,\ldots,n\}$), where $\mathrm{col}_i(A)$ is the $i$th column of $A$ and $\|\cdot\|$ is the Euclidean ... linear-algebra matrices matrix-norms- 581
Finding the positive projection in the frobenius norm
I am trying to find the positive projection in the frobenius norm of a real matrix. Consider the following matrix $\hat{Z}$: \begin{bmatrix}1&2&0\\0&5&0\\0&8&9\end{bmatrix} I ... matrices eigenvalues-eigenvectors matrix-decomposition positive-definite positive-semidefinite- 3
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