Questions tagged [orthogonal-matrices]

Matrices with orthonormalized rows and columns. An orthogonal matrix is an invertible real matrix whose inverse is equal to its transpose. For complex matrices the analogous term is *unitary*, meaning the inverse is equal to its conjugate transpose.

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If $A$ is an orthogonal real matrix that commutes with all orthogonal matrices then $A$ is a scalar multiple of identity matrix.

If $A$ is an orthogonal real matrix that commutes with all orthogonal matrices then $A$ is a scalar multiple of identity matrix. I was able to prove the previous part of the question which is the ...
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Invertibility proof of (I+A) [on hold]

If p(M)< 1 such that (I - M) is invertible, can the same be said for (I+M)?
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Infinitesimal increments to independent variables in orthogonal matrices

I'm trying to read the proof for existence of the Singular Value Decomposition from the Eckart-Young (1936) paper. In page 215, the authors mention "if $u$ is any orthogonal matrix and the ...
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1answer
26 views

Alternative way to show that the special orthogonal group is compact

To show that the special orthogonal group ${\rm SO}(n,\mathbb R)$, carrying the subspace topology induced by ${\rm Mat}_{n}(\mathbb R) \cong {\mathbb R}^{n}$, is compact many proofs use the Heine-...
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Matrix orthogonal if and only if another matrix unitary

For a positive integere $n$, suppose that $C$ and $D$ are real square matrices of size $n$. Let $F$ be defined as $F = C + iD$. And $G$ be defined as $$G=\left[\begin{matrix}C & -D\\D & C\end{...
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For this answer, why is w ∈ V⊥ can also belong to U⊥. Could someone explain this?

If $U \subseteq V$, then $V^\perp \subseteq U^\perp$. In the solution chosen, the answer was given with the assumption that for given any w∈V⊥. But also that w∈U⊥ . Now I can prove that this w ...
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Can the derivative of the matrix absolute value explode when we approach singular matrices?

Let $ \text{GL}^+_n$ be the group of real $n \times n$ matrices with positive determinant, and consider the matrix absolute value function, $| \cdot | : \text{GL}^+_n \to \text{Psym}$ given by $|A|=\...
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124 views

None element of orthogonal matrix can't have unit modulus larger then 1

None element of orthogonal matrix can't have unit modulus larger then 1. I've tried to use the properties of orthogonal matrices ( $|det(A)| = 1$ and $Q^T=Q^{-1}$ ) but I couldn't find out how they ...
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48 views

Infinite-dimensional analogue of orthogonal matrices

In functional analysis one learns that self-adjoint operators are the infinite dimensional generalisation of symmetric matrices and the dual operator is the generalisation of the transposed matrix ...
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Problem on defining the shearing matrice [closed]

How can I define the shearing matrix occurring on (1-10) plane and <111> direction?
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993 views

Confusion on terminology: “vector rejection”, “vector projection”, and “orthogonal projection”?

I have not found a satisfying answer to what is the difference between these three, intuitively: 1) vector projection 2) vector rejection 3) orthogonal projection I know that "vector rejection of ...
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Find the orthogonal projection of y onto subspace of $R^3$ spanned by S, where S is not orthogonal.

y and S are defined below: $$y= \begin{bmatrix} 9 \\ 2 \\ -4 \\ \end{bmatrix} $$ $$S=\{ \begin{bmatrix} 1 \\ 0 \\ 1 \\ \end{bmatrix}, \begin{bmatrix} ...
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Householder matrices for thin QR updating

Suppose you have a QR decomposition of the form: $X=QR$ (Where $X$ is an arbitrary matrix of size $(n \times p)$, $Q$ is an orthogonal matrix of size $(n \times n)$ and $R$ is an upper trapezoidal ...
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What is the physical significance of the determinants of orthogonal matrices having the value of $\pm 1$?

I'm new to linear algebra and while studying orthogonal matrices, I found out that their determinant is always $\pm 1$. Why is that so? What could be the physical significance behind it? I know that ...
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Orthogonal projection matrices to the subspaces ${Ker(X)}$ and ${Ker(X)^{\perp}}$ [closed]

How to determine the orthogonal projection matrices to the subspaces ${Ker(X)}$ and ${Ker(X)^{\perp}}$ , if $Ker(X)=span(v)$, where it is $v≠0$ ?
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19 views

Dimension of the orthogonal algebra?

The following is on page 3 of Introduction to Lie Algebras and Representation Theory by Humphreys: Here the author claims that the dimension of the orthogonal algebra is $2l^2+l$; but I think the ...
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1answer
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General solution using pseudo inverse.

I'm having trouble to understand the general solution of a $Ax=b$ when $x=A^+b+ [I-A^+A]w$ I don't understand why the $w$ is there and why $w$ can be any vector. My view is: $Ax=b$ $A[A^+b+ [I-A^+A]...
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2answers
47 views

Orthogonal matrices and matrix norms

I have seen some disagreement online and was wondering if anyone could clarify for me: If $X$ is an arbitrary $n \times n$ matrix and $A$ is an arbitrary orthogonal $n \times n$ matrix, is it true ...
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2k views

Orthogonal matrix over cross product

Let $a$ and $b$ be two unitary vectors in $\mathbb E^3$, and let $Q$ be an orthogonal matrix. Does the following hold? $$Qa \wedge Qb = \pm Q(a \wedge b)$$
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51 views

Irreducible representations of $SO(2)$ on 2x2 matrices.

I'm having trouble verifying my understanding of the representation theory of Lie groups (which is minimal) with my experience playing around with the rotations of 2x2 matrices. Specifically, if we ...
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1answer
50 views

A Question on the existence of an orthogonal matrix in Linear Algebra

The following is an exercise in my linear algebra textbook. Suppose $\vec{x} \in \mathbb{R}^n$ and $\|\vec{x}\|^2 = 1$. Prove that there exists a matrix $A \in O(n,\mathbb{R})$ and $A^T=A$ such ...
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44 views

Identifying a “rotated shear” matrix

Suppose that we're in $\Bbb R^n$. Then the simplest shear matrix can be described as $$S_\lambda = \begin{bmatrix} 1 & \lambda & 0 & \ldots & 0 \\ 0 & 1 & 0 & \ldots &...
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Is sum of two orthogonal matrices singular?

I am trying to solve following problem. Let $A, B \in \mathbb{R}^{n\times n}$ be an orthogonal matrices and $\det(A) = -\det(B)$. How can it be proven that $A+B$ is singular? I could start with ...
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Difference between orthogonal and orthonormal matrices

Let $Q$ be an $N \times N$ unitary matrix (its columns are orthonormal). Since $Q$ is unitary, it would preserve the norm of any vector $X$, i.e., $\|QX\|^2 = \|X\|^2$. My confusion comes when the ...
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2answers
759 views

Uniqueness of orthonormal basis

i computed the orth of a matrix on matlab: Why is it different from http://www.wolframalpha.com/input/?i=orthogonalize+%7B(1,+i,+2-i,-1),+(2%2B3i,+3i,+1-i,+2i),+(-1%2B7i,+6%2B10i,+11-4i,+3%2B4i)%7D?
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169 views

On matrix tridiagonalization

Can any matrix $X \in \mathbb R^{n \times n}$ be decomposed into a tridiagonal matrix, i.e., $$X = P^{-1}DP$$ where $P \in \mbox{SO}(n)$ and $D$ is tridiagonal?
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find power of orthonormal matrix to get identity matrix

I have this orthonormal matrix: $ Q =\frac{1}{9} \left(\begin{matrix} 4 & 8 & -1 \\ -4 & 1 & -8 \\ -7 & 4 & 4 \end{matrix}\right)$ If I calculate $Q^4$, I get the identity ...
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Translation as product of reflections

I am facing the following problem, given the translation in the euclidean affine space of dimension 4 $ \tau_v= \begin{bmatrix} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ ...
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1answer
52 views

Haar measure from axis-angle representation of $SO(3)$

My aim is to study some special representations of $SO(3)$ using characters, and to do this I need an explicit expression of the Haar measure on $SO(3)$. I've found some "versions" of the Haar measure ...
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31 views

What's about the sum of two orthogonal vectors

I'd like to ask if the sum of two orthogonal vectors is a vector which is orthogonal on others, where can I get more details about that? for example, suppose we have the Walsh matrix of 4, which is ...
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83 views

Can an orthogonal non-symmetric 3x3 matrix have 3 real eigenvalues?

I was wondering if a non-symmetric orthogonal matrix can have his 3 eigenvalues in the real numbers. All the 3 real eigenvalues orthogonal matrix i've found are symmetric. Can someone give me a ...
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1answer
55 views

Conjugation of $\left(\begin{smallmatrix} 0 & -1 \\ 1 & 0 \end{smallmatrix}\right)$

I'm interested in the following question. Let $h=\begin{pmatrix} 0 & -1 \\ 1 & 0 \\ \end{pmatrix}$. This is an orthogonal map which is quite far away from the identity (say in the Frobenius ...
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(real symmetric matrices) does orthogonality of eigenvectors (distinct eigenvalues) depend on choice of basis?

For some reason I cannot wrap my head around this one. My schoolbook begins the chapter on eigendecomposition of real symmetric matrices by stating that eigenvectors from distinct eigenspaces are ...
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47 views

Orthogonal diagonalization without eigenvectors

I stumbled onto a method for orthogonally diagonalizing a symmetric matrix with real entries and I was wondering what advantages (if any at all) it has over the eigenvector method. It hinges on the ...
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1answer
27 views

Calculate matrix $A^T A$ with pairwise orthogonal vectors

I have a matrix $A$, that contains pairwise orthogonal vectors with length $1$, and I should calculate $A^T A$. I defined that: $ v_{1}, v_{2}, v_{n-1}, v_{n} ∈ R^n \ and \ A ∈ R^{m _x n} $ and if I ...
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Trouble understanding orthonormal vectors.

I am pretty sure the definition of orthonormal matrix is a matrix whos columns contain vectors that are orthogonal to each other and all of length 1. In that case I cant understand 17:04 of this ...
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Induction step of proof of 'Every element of $O(n)$ is product of hyperplane reflections'

The following came up in induction step of proof of 'Every element of $O(n)$ is product of hyperplane reflections'. An element $A$ in $O(n)$ is called hyperplane reflection if $$A=Pdiag(1,\cdots , 1,-...
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Find eigenvalues, kernel and Image of an Orthogonal projection

Let $V$ be a Vector space with inner product and $U$ a subspace. Let $P$ be the orthogonal projection over $U$. Find eigenvalues, kernel and Image of $P$. I know I have to consider the special cases ...
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What are the properties of eigenvalues of permutation matrices?

Up till now, the only things I was able to come up/prove are the following properties: $\prod\lambda_i = \pm 1$ $ 0 \leq \sum \lambda_i \leq n$, where $n$ is the size of the matrix eigenvalues of the ...
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Orthogonal Complement of the Column Space [closed]

I currently have this problem with this matrix. Of this matrix i have to calculate the Orthogonal Complement of the Column Space. But nothing is given? How can you do this? Thank you in advantage.
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Are the derivatives of the orthogonal polar factor locally integrable?

Let $\mathbb{D}^2$ be the closed unit disk, and let $f:\mathbb{D}^2 \to \mathbb{R}^2$ be a real-analytic map, satisfying $\det df>0$ everywhere except on a set of Hausdorff dimension not greater ...
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281 views

How many degrees of freedom do orthogonal skew-symmetric matrices have?

$n$ by $n$ real orthogonal matrices have $n (n-1)/2$ degrees of freedom. So do the skew-symmetric matrices. But what about matrices that are both skew-symmetric and orthogonal? Is the number of such ...
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Generate orthogonal (lower) upper-triangular matrices

Problem I am trying to numerically verify the fact that "the orthogonal lower (upper) triangular matrix has to be diagonal". However, I have difficulty finding general matrices that satisfies both ...
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Then which of the following statements are true?[CSIR-2018-December]

Let $\{u_1,u_2,..., u_n\}$ be an orthonormal basis of $\mathbb {C^n}$ as column vectors. Let $M=(u_1,u_2,...,u_{k})$ and $N=(u_{k+1},u_{k+2},...,u_{n})$ and $P$ be a $k \times k$ diagonal matrix with ...
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1answer
799 views

Orthogonal Matrix with Determinant 1 is a Rotation Matrix

I am confused with how to show that an orthogonal matrix with determinant 1 must always be a rotation matrix. My approach to proving this was to take a general matrix $\begin{bmatrix}a&b \\c&...
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Find a unitary basis of the $\mathbb{R}$-vector space of $n \times n$ (complex) Hermitian matrices

The question is on the title ($n \in \mathbb{N}^*$). To be clear, unitary basis here means basis consisting of (complex) unitary matrices. I wonder this question because recently I've read about ...
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Eigenvalues in orthogonal matrices

Let $A \in M_n(\Bbb R)$. How can I prove, that 1) if $ \forall {b \in \Bbb R^n}, b^{t}Ab>0$, then all eigenvalues $>0$. 2) if $A$ is orthogonal, then all eigenvalues are equal to $-1$ or $...
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1answer
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constructive canonical form of orthogonal matrix

For every orthogonal matrix $Q$ over the reals there is an orthogonal matrix $P$ and a block diagonal matrix $D$ such that $D=PQP^{t}$. Each block in D is either $(1)$, $(-1)$ or a two dimensional ...
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Why does $A^TA=I, \det A=1$ mean $A$ is a rotation matrix?

I know if $A^TA=I$, $A$ is an orthogonal matrix. Orthogonal matrices also contain two different types: if $\det A=1$, $A$ is a rotation matrix; if $\det A=-1$, $A$ is a reflection matrix. My question ...
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Is relationship of orthonormality & orthogonality equivalent to that of squares & rectangles?

To confirm my analogy, I am asking if I can consider every orthonormal basis to be an orthogonal one (but not vv) in the same sense that every square is a rectangle (but not vv). I believe the answer ...