# Tagged Questions

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### Identities for the Hilbertâ€“Schmidt norm of products of projections.

I've been studying different metrics on the Grassmannian $Gr(k,n)$ of k-dimensional linear subspaces of $\mathbb{R}^n$ and found myself needing some identities for the norm of a product of orthogonal ...
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### Normalize matrix so that its projection equals identity - which to use?

I wish to normalize a given matrix M, n by k, so that its projection matrix equals the identity. This is to speed up computations. The transformation/decomposition would look like this: Generate M by ...
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### Left-invariant Riemannian metric on $SO(3)$

Let's consier the manifold $SO(3)$. First problem is to show that $T_I SO(3)$ is a space of skew-symmetric matrices $3\times 3$. How can I deduce it? Then I have to prove there exists exactly one ...
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### Does R' = R*R'*R^(-1) hold?

Given two 3D rotation matrices ($3\times 3$) $R$ and $R'$, does this equivalence hold?: $R' = R*R'*R^{-1}$ My intuition tells me so, but I can't find a formal proof for it. Thanks.
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### Orthonormal matrix by diagonal matrix by orthonormal matrix transpose. [closed]

Let $D$ be a diagonal $n \times n$ matrix and $V$ an $n\times n$ matrix whose columns are orthonormal. Is it true that $$VDV^T = D?$$ If so, why?
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### Inner product space, orthonormal bases and change of basis.

I define unitary as $B*B=I$ I know that part (i) requires me to show the matrix coefficients are that of the inner product for bases A and B, however I am unsure how to get to this. Any help would ...
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### Choosing an orthonormal basis in which a linear operator has a sparse matrix

Given a linear operator $T$ on an $n$-dimensional vector space $V$ (over $\mathbb R$), I want to find an orthonormal basis for $V$ in which the matrix of $T$ is sparse (has many zeros). How many zeros ...
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### Is there a simple method to finding orthonormal basis given a partially complete set

I have a question Find the indicated projection matrix for the given subspace, and find the projection of the indicated vector $<2,-1,3>$ on ...
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### Inverse of orthogonal matrix is orthogonal matrix?

Is inverse of an orthogonal matrix an orthogonal matrix? I know its inverse is equal to its transpose, but I don't see where the orthogonality would come from.
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### bounding operator norm by quadratic forms on orthonormal basis

Let $\{u_i\}_{i=1}^n$ be a set of orthonormal basis and $M$ a symmetric matrix. Suppose $$\left|\langle u_iu_i^T,M\rangle\right|\leq \tau\ \ \forall i=1,\dots,n.$$ Can I ...
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### How to find a suitable orthogonal matrix

Assume $x,y \in \text{R}^n$ are two vectors of the same length, how to prove that there is an orthogonal matrix $A$ such that $Ax=y$? Thanks for your help.
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### Determine the matrices that represent the following rotations of $\mathbb{R}^3$

I need to determine the matrix that represents the following rotation of $\mathbb{R}^3$. (a) angle $\theta$, the axis $e_2$ (b) angle $2\pi/3$, axis contains the vector $(1,1,1)^t$ ...
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### Change of basis (Gram-Schmidt)

I was wondering whether it is possible to write down explicitely the matrix that represents the change of basis from a basis $\{v_1,....,v_n\}$ to a basis $\{e_1,...,e_n\}$, where $e_i$ is the basis ...
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### 'Interesting' orthogonal martices?

I'm working with an algorithm that takes as part of its input an orthogonal matrix with real entries. I'm trying to understand how the specification of this matrix affects performance of the ...
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### Is the largest number of zero eigenvalues at least 2

For any $x\le3$, is the largest (or maximal) number of zero eigenvalues of $diag\left(1,2,3\right)-U\cdot diag\left(4,5,x\right)\cdot U^{T}$ for orthogonal matrix $U$ at least $2$?
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### Prove that for an Hermitian matrix $A$ exists one and only matrix $S\in \text{GL}(\mathbb{C},n)$ such that …

The task is to prove that for an Hermitian matrix $A$ exists one and only matrix $S\in \text{GL}(\mathbb{C},n)$ upper triangular with positive entries on diagonal such that $A = S^T \bar{S}$? As I ...
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### Finding Matrix of improper rotation

I'm struggling with this excercise: Give a Matrix $A\in O(3)$ which is describing an improper rotation with an angle of $\pi /3$ and axis $(1,1,1)$. What do I need to do?
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### How to calculate an orthonormal basis for a matrix?

Are there any specific, easy to compute, algorithms to build an orthonormal basis for a matrix in which each column has length one?
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### Efficient (approximate) projection onto the special orthogonal group

I need to carry out an optimization on the special orthogonal group $SO(n)$. For the line search I use a simple back-projection method $$\mbox{minimize}_\tau f(\pi(X+\tau Z))$$ where $X\in SO(n)$ ...
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### Gram-Schmidt verifying orthonormal basis

Gram-Schmidt If I have an orthonormal basis, how do I verify that they are indeed orthonormal? I have Q, R and A is it enough to times Q` by Q to give me I? or A=QR? Edit: Let's say I ...
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### What's the trick for proving one eigenvalue of orthogonal matrix is $-1$ if the determinant is $-1$?

Obviously, the magnitude of the orthogonal matrix is 1, which is easy to prove.. However, I wonder how can one prove that the eigenvalue of an orthogonal matrix is $-1$, if the determinant of this ...
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### How to prove unitary matrices require orthonormal basis

How to prove unitary matrices require orthonormal basis thanks much!
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