0
votes
0answers
16 views

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 ...
1
vote
1answer
45 views

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 ...
0
votes
3answers
58 views

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.
0
votes
1answer
32 views

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?
0
votes
1answer
64 views

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 ...
2
votes
0answers
48 views

Calculating the signature of a matrix

The task is the following: Consider $\mathbb{R}^2$ equipped with the canonical dot-product $\langle \cdot , \cdot \rangle$, and also the symmetrical bilinear form $$\beta(u,v) := \left\langle u,\ ...
1
vote
2answers
23 views

orthonormal vector properties

I have noticed a matrix property that is outlined below: I have a set of n orthonormal eigenvectors that form a basis in Rn. If these vectors are combined to form an nxn matrix where each column is ...
1
vote
1answer
93 views

Dot product with vector and its transpose?

I'm having trouble with the statement: $$||\textbf{v}||^2=\textbf{v}\cdot\textbf{v}=\textbf{v}^T\textbf{v}$$ taking $\textbf{v}$ as a column vector in an orthogonal matrix. How can you do the dot ...
1
vote
1answer
34 views

Is every unit vector in $\mathbb{Q}^n$ the first column of a rational orthogonal matrix?

Equivalently, does every unit vector in $\mathbb{Q}^n$ belong to some orthonormal basis for $\mathbb{Q}^n$? This is clearly true for $\mathbb{Q}^2$, and for $\mathbb{Q}^3$ it seems to be true for ...
1
vote
1answer
46 views

Name for multiples of orthogonal matrices

Is there a name for a matrix which is a multiple of an orthogonal matrix? I.e. a square matrix $A$ which satisfies the condition $$A^TA = AA^T = \lambda I$$ where $\lambda$ is some scalar (which ...
4
votes
2answers
42 views

Finding the dimension and basis of an orthogonal space

I am trying to find the basis and dimensions of the the space orthogonal $S$ which is in $\mathbb R^3$. $$S = \begin{bmatrix}1\\2\\3\end{bmatrix}$$ So the dimension would be two because it is $3 - ...
1
vote
1answer
108 views

Eigenvalues of a $3\times3$ orthogonal matrix

Can anyone give me an example of 3x3 orthogonal matrix with complex eigenvalue.
1
vote
0answers
61 views

Will the projection of a singular matrix into an orthonormal space be non-singular?

I'm working through an implementation of the solution from 16.3.1 Dealing with the nullspace in the case of a singular within-class scatter matrix when performing discriminant analysis. In this ...
12
votes
3answers
474 views

Why is an orthogonal matrix called orthogonal?

I know a square matrix is called orthogonal if its rows (and columns) are pairwise orthonormal But is there a deeper reason for this, or is it only an historical reason? I find it is very confusing ...
0
votes
1answer
26 views

relationship between matrix and adjoint of a matrix and orthonormal system

The problem is: Let $A$ be a $m\times n$ matrix. Show that if $A^*A=I$, the $n\times n$ identity matrix ($A^*$ is the adjoint), then the columns $g_1,\cdots,g_n$ in $A$ constitute an orthonormal ...
0
votes
1answer
83 views

Question about Gram-Schmidt algorithm. Orthogonal diagonalization. Does GS conserve eigen-ness property

I have a question about the Gram-Schmidt process, and about the algorithm to find an orthogonal basis of eigenvectors (aka orthogonal diagonlization). let $T:V \to V$ be a diagonlizable linear ...
3
votes
0answers
91 views

Question about orthogonal transformation / orthogonal matrices

I have a question about orthogonal transformations. If $T$ is an orthogonal transformation from $V$ to $V$, should the representation matrix with respect to any orthonormal basis of any inner product ...
2
votes
3answers
112 views

$A$ is a symmetric real matrix. Show that there is $B$ such that $B^3=A$

I'm having trouble with this question, I'd like someone to point me in the right direction. let $A$ be a n by n matrix with real values. show that there is another n by n real matrix $B$ such that ...
1
vote
2answers
35 views

Finding a line of approximation using the normal equations for $A\vec{x}=\vec{b}$

To find the line $y=ax+b$ that best approximates the data points $\{(-2,3),(0,5),(1,7)\}$ I need to use the equation $$A\vec{x}=\vec{b}\ \ \ (\mbox{where}\ \vec{x}=\left({a\atop b}\right))$$ Then ...
1
vote
4answers
95 views

Proving the standard matrix U of T to be orthogonal

So my class is getting into orthogonality, however, our reading assignments haven't been touching on transformations. I have this proof problem that I cannot seem to get around. Does anyone have any ...
0
votes
1answer
118 views

matrices in the gram schmidt process

I have a question that says: Use the Gram-Schmidt orthonormalization process to transform the given basis for a subspace into an orthonormal basis for the subspace. Be sure to show your matrix ...
0
votes
1answer
118 views

How to prove $det(A) = 1$ or $-1 \Longrightarrow AA^t = A^tA = I_n$?

Prove $det(A) = 1$ or $-1 \Longrightarrow AA^t = A^tA = I_n$? I have no clue, to be fair. I am trying to prove orthogonal polynomials have a det = 1 - help?
3
votes
1answer
59 views

Orthogonal $M$ such that $M^TAM$ is upper triangular?

We triangularise $$A=\begin{bmatrix}2&1&1\\-20&-5&-10\\16&3&8 \end{bmatrix}$$ into $U=S^{-1}AS$ with $$S=\begin{bmatrix}-1&0&0\\0&-1&0\\2&1&1 ...
13
votes
4answers
386 views

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 ...
0
votes
1answer
60 views

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 ...
5
votes
3answers
2k views

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.
0
votes
1answer
67 views

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 \begin{equation} \left|\langle u_iu_i^T,M\rangle\right|\leq \tau\ \ \forall i=1,\dots,n. \end{equation} Can I ...
2
votes
3answers
74 views

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.
2
votes
1answer
167 views

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 $(1,1,1)^t$ ...
1
vote
2answers
83 views

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 ...
1
vote
0answers
70 views

'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 ...
1
vote
1answer
44 views

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 $2$?
1
vote
1answer
70 views

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 ...
0
votes
1answer
129 views

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?
0
votes
0answers
681 views

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?
0
votes
1answer
90 views

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 $X\in SO(n)$ ...
0
votes
2answers
149 views

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 ...
2
votes
1answer
1k views

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 ...
2
votes
1answer
317 views

How to prove unitary matrices require orthonormal basis

How to prove unitary matrices require orthonormal basis thanks much!
1
vote
1answer
24 views

A Problem with a Proof Concerning the Properties of Orthogonal Matricies

OK, I'm working on the following problem and don't understand how my Linear Algebra text has made it to a certain conclusion here is the problem: Let u be a unit vector in $R^{n}$ and let $H = I - ...
1
vote
2answers
320 views

Finding all orthogonal matrices of a given form

I am given this problem. I need to find all $a,b,c,d \in \mathbb{R}$ for which the matrix $A$ is orthogonal. $$A:=\frac{1}{3} \begin{pmatrix} -1 & 2 & a \\ 2 & 2 & b \\ 2 & c ...
3
votes
4answers
513 views

Normalizing a matrix

I came across a step in an numerical algebra algorithm that says "Normalize the rows of matrix A such that they are unit-norm. Call U the normalized matrix." I do something like this: ...
2
votes
1answer
76 views

Is there a matrix satisfying a certain condition

I have a numerical problem which boils down to the following: We are given a square matrix $R$, with a bunch of zeros in it. We want to check if there exists orthonormal matrix $T$ such that $TT'=I$, ...
1
vote
1answer
89 views

how to compute matrix polar decomposition of a non square matrix

I know that for any matrix $A\in \mathbb{C}^{n\times m}$ we can find a polar decomposition such that $A=UP=U|A|=U\sqrt{A^*A}$ where $P\in \mathbb{C}^{m\times m}$ and $U\in \mathbb{C}^{n\times m}$ is ...
1
vote
1answer
177 views

Verifying Orthogonality of Eigenvectors

How do you 'verify' the orthogonality of the eigenvectors of a matrix, let's say ${\bf A}$ , for example? I came across the result that a matrix ${\bf A}$ has orthogonal eigenvectors if ${\bf ...
0
votes
1answer
202 views

self-adjoint operator and unitary orthogonal matrix

Please offer a solution to the following problem. It was offered in class by my professor as an additional exercise to try on one's own. Let $V$ be the inner product space, and assume that $\alpha ...
1
vote
2answers
536 views

Orthonormalization

I have a math problem I am struggling with: If a linear transform $A: \mathbb{R}^n\to\mathbb{R}^n$ and we have a basis of $\mathbb{R}^n$ of eigenvectors of $A$, can't we just orthonormalize them and ...
0
votes
1answer
293 views

Orthonormalization of non-hermitian matrix eigenvectors

I have been told that orthonormalization of the eigenvectors of a non-hermitian matrix has to use a different definition of inner product than when the matrix is hermitian. Why is this so, and how do ...
2
votes
1answer
2k views

How to find the orthonormal transformation that will rotate a vector to the x axis?

I am having trouble remembering linear algebra. I need to find the orthonormal transformation that will rotate a 3-dimensional vector to the x axis. I could not find any similar question on the net. ...
2
votes
1answer
102 views

Diagonalizing a matrix for producing a orthogonal set of functions.

Given a basis $B = \{g_0, g_1,\ldots , g_{nk}\} $, I wish to construct a set of $nk$ functions $S$ such that $\langle S_i,S_j \rangle = \delta_{i,j}$ (so that $S$ is an orthonormal set) where $\langle ...