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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### What is the geometric interpretation of the transpose?

I can follow the definition of the transpose algebraically, i.e. as a reflection of a matrix across its diagonal, or in terms of dual spaces, but I lack any sort of geometric understanding of the transpose, or even symmetric matrices.
For example,…

Zed

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### In which cases is the inverse of a matrix equal to its transpose?

In which cases is the inverse of a matrix equal to its transpose, that is, when do we have $A^{-1} = A^{T}$? Is it when $A$ is orthogonal?

Chris

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### Why is the matrix product of 2 orthogonal matrices also an orthogonal matrix?

I've seen the statement "The matrix product of two orthogonal matrices is another orthogonal matrix. " on Wolfram's website but haven't seen any proof online as to why this is true. By orthogonal matrix, I mean an $n \times n$ matrix with…

a12345

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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 $1$

Kuba

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### Why are orthogonal matrices generalizations of rotations and reflections?

I recently took linear algebra course, all that I learned about orthogonal matrix is that Q transposed is Q inverse, and therefore it has a nice computational property.
Recently, to my surprise, I learned that transformations by orthogonal matrices…

Alby

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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 columns of $Q$ are orthogonal, but not orthonormal,…

user63552

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### Is the seven-dimensional cross product unique?

I'm confused about how many different 7D cross products there are. I'm defining a 7D cross product to be any bilinear map $V \times V \to V$ (where $V$ is the inner product space $\mathbb{R}^7$ endowed with the Euclidean inner product) such that for…

tparker

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### Why is the orthogonal group $\operatorname{O}(2n,\mathbb R)$ not the direct product of $\operatorname{SO}(2n, \mathbb R)$ and $\mathbb Z_2$?

We know that when $n$ is odd, $\operatorname{O}_n(\mathbb R) \simeq \operatorname{SO}_n (\mathbb R) \times \mathbb Z_2$.
However, this seems not true when $n$ is even. But I have no idea how to prove something is not a direct product.
I have tried…

Roun

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### Sub-determinants of an orthogonal matrix

Let $A$ be a matrix in the special orthogonal group, $A \in \mathrm{SO}_n$. This means that $A$ is real, $n \times n$, $A^t A = I$ and $\det(A)=1$, that is, the column vectors of $A$ make a positively-oriented orthonormal basis for $\mathbb…

Ryan Budney

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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 is: what is the relationship between the…

Shiyu

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### Is it possible to have a $3 \times 3$ matrix that is both orthogonal and skew-symmetric?

Is it possible to have a $3 \times 3$ matrix that is both orthogonal and skew-symmetric?
I know it has something to do with the odd order of the matrix and it is not possible to have such a matrix. But what is the reason?

sagar bangal

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### Does SO(3) preserve the cross product?

Let $g\in SO(3)$ and $p,q\in\mathbb{R}^3$. I wondered whether it is true that $$g(p\times q)=gp\times gq$$
I am not sure how to prove this. I guess I will use at some point that the last row $g_3$ of $g$ can be obtained by $g_3=g_1\times g_2$.
But I…

Valentin

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### Orthogonal matrices form a compact set

Prove that the set of all $n \times n$ orthogonal matrices is a compact subset of $\mathbb{R}^{n^2}$.
I don't know how it can be done. Thanks.

Pedro

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### Eigenvalues of symmetric orthogonal matrix

Can we say that Eigenvalues of symmetric orthogonal matrix must be $+1$ and $-1$?
Since eigenvalues of symmetric matrices are real and eigenvalues of orthogonal matrix have unit modulus. Combining both result eigenvalues of symmetric orthogonal…

mathscrazy

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### Which matrices $A\in\text{Mat}_{n\times n}(\mathbb{K})$ are orthogonally diagonalizable over $\mathbb{K}$?

Update 1. I still need help with Question 1, Question 2' (as well as the bonus question under Question 2'), and Question 3'.
Update 2. I believe that all questions have been answered if $\mathbb{K}$ is of characteristic not equal to $2$. The only…

Batominovski

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