1
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1answer
29 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
41 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
40 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
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1answer
81 views

Eigenvalues of a $3\times3$ orthogonal matrix

Can anyone give me an example of 3x3 orthogonal matrix with complex eigenvalue.
1
vote
0answers
47 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 ...
11
votes
0answers
272 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
18 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
61 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
74 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
71 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
34 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
72 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
84 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
104 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
46 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
306 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
40 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
1k 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
55 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
69 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
153 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
72 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
65 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
42 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
69 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
101 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
81 views

Orthogonalising the standard finite element hat function basis - Mass matrix

If one wants to find the $L^2$ projection of a function f the standard finite element space $V_n$ spanned by basis functions $\{\varphi_i\}_{i=1}^N$, then you solve $A\alpha=\beta$ where ...
0
votes
0answers
585 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
0answers
43 views

How to calculate orthogonal projection of one dimension vector

refer to http://mathoverflow.net/questions/60185/linear-combination-of-orthogonal-projection-matrices if use one dimension vector to calculate orthnormal basis by Gram-Schmidt algorithm. then how to ...
0
votes
1answer
80 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
0answers
24 views

when gives svd decomposition matrices,whose element are different from 0 and 1

i want to understand one basic question,what is a criteria that for given matrix,whose element are just only $1$ and $0$,it's svd decomposition gives me matrices, whose element are different from ...
0
votes
2answers
128 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 ...
0
votes
1answer
255 views

How to prove unitary matrices require orthonormal basis

How to prove unitary matrices require orthonormal basis thanks much!
1
vote
1answer
23 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
225 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
491 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
79 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
162 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
182 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
456 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
276 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
1k 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
101 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 ...
2
votes
2answers
320 views

Can all matrices be orthogonalized?

I helped a buddy of mine do some MATLAB homework where you had to orthogonalize a matrix via Gram-Schmidt. I wrote a test function called isorthog that returned ...
5
votes
3answers
431 views

Freedoms of real orthogonal matrices

I was trying to figure out, how many degrees of freedoms a $n\times n$-orthogonal matrix posses.The easiest way to determine that seems to be the fact that the matrix exponential of an antisymmetric ...
1
vote
1answer
214 views

Rotation matrices for arbitrary dimensions

I initially asked this question here, and someone suggested this may be a better place to get an answer. I have a question about a rotation matrix, which can be represented in 2 dimensions as: ...
2
votes
6answers
298 views

determining orthonormal matrix of rank N with special first row

Is there a more efficient algorithm besides Gram-Schmidt that would produce an orthonormal matrix of rank N, with first row equal to [1 1 1 1 1 ... 1] / sqrt(N)? e.g. for N = 3, the matrix ...