# 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.

670 questions
Filter by
Sorted by
Tagged with
22 views

### Converting a linear space into a orthonormal base

I've been given a linear space made by 2 equations $U = \{\ {(x,y,z,w) ∈ R^4: x - y + z = 0 \land y - z + w = 0} \}\$ and I have to get the orthonormal base of this equation and what I did to to that ...
24 views

### Distance of a vector from a subspace made by equations in Linear Algebra

So I have an exercise which gives my me this subspace : $U = {(x,y,z) ∈ R^3: x - y - z = 0 \text{ and } x + y + z = 0}$ and i have to determine the orthogonal projection of $(3,0,-1)$ which i did, ...
26 views

55 views

### Why is every element in $SO(3)$ is a rotation?

Question: I am worried about the proof of Equivalence of an orthogonal matrix to a rotation matrix provided in this Wikipedia page, for the change of basis matrix may be a complex but not real matrix, ...
43 views

19 views

46 views

### determinant of an orthogonal matrix

The question goes like this, For a square matrix A of order 12345, if det(A)=1 and AA'=I (A' is the transpose of A) then det(A-I)=0 (I have to prove it if it is correct and provide a counterexample ...
46 views

36 views

26 views

### How can I show that every vector in $U\oplus W$ can be uniquely decomposed into a sum of vectors in $U$ and $W$?

How can I show that every vector in $U\oplus W$ can be uniquely decomposed into a sum of vectors in $U$ and $W$? Let $V$be an inner product space over $\mathbb{R}$ (here, the inner product is denoted ...
36 views

20 views

### Obtaining a orthonormal matrix with 2 columns given

I have to create a orthonormal square matrix whose first row and last row, say $T_1$ and $T_N$ are given. Now what should be done to get the remaining columns of the matrix? The Gram-Schmidt ...
63 views

### A curious interrelationship between distinct embeddings of $SO(M+1)$ into $SO(2M+1)$

The following seems to be a property of $SO(2M+1)$ for an arbitrary integer $M$, although I have not yet been able to prove it. (I can prove it for, e.g., $M=1$, and have numerically checked it for ...
For each $\ n \in \mathbb{N} \$ let $\, O(n+1) \,$ be the set of orthogonal linear operators $\ \mathbb{R}^{n+1} \to \mathbb{R}^{n+1} \,$, $\ e_{n+1} = (1,0,0,...,0) \in \mathbb{R}^{n+1} \$ ...