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.

# Questions tagged [orthogonal-matrices]

961 questions

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### Is there a name for matrices with singular values all equal to $1$?

I at first thought these are just the orthogonal matrices, since the SVD of $X$ is $U \Sigma V^T$ and if $\Sigma$ is the identity matrix then this implies X is orthogonal.
However, singular values equal to 1 doesnt imply $\Sigma$ is identity,…

user56834

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### Let A be an $n \times k$ matrix with $W$ as the column space. Show that the solution space of $A(A^T) x = 0$ is given by $W^⊥$.

How can I solve this question? $A^T$ refers to $A$ transposed and $W^⊥$ refers to the set of vectors orthogonal to W.

LolOLOL

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### A question related to skew symmetric matrix and Orthogonal matrices

Consider the following problem asked in a masters exam for which I am self studying.
Write V for the space of $3 \times 3$ skew - symmetric real matrices.
(A) Show that for $A\in SO_3(\mathbb{R})$ and $M\in V$ , $AMA^t \in V$.
(B) Show that…

Avenger

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### If a square matrix's column vectors and row vectors all have norm 1, then is the matrix orthonormal?

Well, this is a question I had to ask myself while solving a problem that asked me to prove a matrix is orthonormal.
I could show that both the column vectors and the row vectors of said matrix all had unit length, but didn't know how to proceed…

Francisco

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### Does $\sum_i e_i \otimes e_i$ in $V\otimes V$ have a name?

If $V$ is a vector space with orthogonal basis $(e_i)$, consider the element
$\sum_i e_i \otimes e_i \in V\otimes V$.
It can be shown that this element is independent of the choice of orthogonal basis.
Does this element have a name?

Hausdorff

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### A continous extension of the group of permutation matrices

For any $n \in \mathbb{N}$, the collection $\mathcal{P}_n$ of permutation matrices is obtained by permuting the rows of the $n \times n$ identity matrix. Under the operation of matrix multiplication, $\mathcal{P_n}$ is a subgroup of $\mathcal{G_n}$…

Miheer

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### An orthogonal matrix in $\mathbb{R}^{3\times3}$ with real eigenvalues is diagonalizable

I know there are two non trivial (i.e. if we solve these two cases the other cases are trivial) cases:
$\lambda_{1,2,3}=1$
and:
$\lambda_1=1,\lambda_{1,2}=-1$
I have been trying to use generalized eigenvectors and the Jordan Canonical and the fact…

Christopher Quinn La Fond Jr.

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### $3\times 3$ orthogonal matrix, which doesn't consist of zeros and ones

I'm stuck with my homework in a subject called Matrices in Statistics. Can you guys help with the following task? I would be very thankful!
The task is as follows:
Find a $3\times 3$ orthogonal matrix, which doesn't consist of zeros and ones.
A…

M. Smithy

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### Is this statement true $AA^t = A^{-1}$

I want to know if its true or not.
According to what I have read this is true for orthogonal matrices. Is it true or not ?
Are there any other cases in which this could be true?

lemniscate

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### Given an orthogonal matrix, how can I find a second orthogonal matrix that gives a product of zero?

Assume I have some orthogonal matrix $\mathbf{Q}^\text{T}\mathbf{Q} = \mathbf{I}$, How can I find a second orthogonal matrix $\mathbf{S}^\text{T}\mathbf{S} = \mathbf{I}$ that gives a product of zero, i.e
$$
\mathbf{Q}^\text{T}\mathbf{S} =…

user2350366

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### Prove that $\|Q\mathbf{v}\|=\|\mathbf{v}\|$

Prove that if $Q$ is a real $n\times n$ orthogonal matrix and $\mathbf{v}$ is in $\mathbb{R}^{n}$, then $$\|Q\mathbf{v}\| = \|\mathbf{v}\|.$$ Be sure to set out your arguments clearly and logically, giving full reasons.
Hello all,
To solve this…

Luke Xu

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### Proving Properties of orthogonal Matrix

I was given a task whereby its defined that a $n \times n$ matrix, A, is orthogonal if $\langle A\vec{u},A\vec{v}\rangle$ = $\langle \vec{u}$,$\vec{v}\rangle$ and i have also been given the property that
$\langle A\vec{u},\vec{v}\rangle$ = $\langle…

user834982

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### Proof for one of the properties of Orthogonal matrices

Consider a matrix O, let's assume it has orthonormal basis. If this...
$$o^{T}=o^{-1}$$ is satisfied, then 0 is a orthogonal matrix. But how does one go to prove that the inverse of an orthogonal matrix is equal to its transpose?
(Basically can…

EPIC Tube HD

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### If matrix $A$ is unitary and $B^2 = A$ then $B$ is also unitary

I need to prove or give a counterexample:
If matrix $A$ is unitary and $B^2 = A$ then $B$ is also unitary
I think the statement is true since the unitary matrix A can only be Identity matrix I or negative identity matrix $-I$; and $B=A^2$ is an…

Denny Shen

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### Gram-Schmidt orthogonalization: Dealing with Complex numbers

A complex valued matrix "A" has n columns a_1 through a_n. Elements of these columns are complex numbers.
The orthogonal complex valued matrix U of A has n columns as well u_1 through u_n.
u_1 is same as a_1.
Projection of column a_x on u_y is…

Raj

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