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]
1123 questions
3
votes
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Why is orthogonal projection matrix has property that $P=P^T$
I know that definition of orthogonal projection matrix is
its range and null space are orthogonal.
And, definition of projection matrix is $P=P^2$
Then, I understand that if $P$ is $n\times n$ projection matrix.
$P=P^T \Rightarrow P$ is orthogonal…
2
votes
0 answers
The terminology of orthogonal matrix vs. orthonormal matrix
I know the definition of an orthogonal matrix. Also there are orthonormal basis. And if two vectors are orthogonal, their scalar product is zero etc. It seems the name "orthonormal matrix" more relevant to other definitions. My question is why it is…
nlite
- 33
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Does the following unitary matrix factorization have a name?
I know any unitary matrix can be factored as follows:
$$\underline {\overline {\bf{U}} } = \left( {\prod\limits_{j = N}^1 {\underline {\overline {\bf{\Psi }} } \left( {{{\underline {\bf{w}} }_j}} \right)} } \right)\underline {\overline {\bf{\Phi }}…
John Polcari
- 640
2
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3 answers
If an orthogonal matrix has determinant -1 then it has -1 as an eigenvalue
I want to show that if a real orthogonal matrix $A$ has determinant $-1$ then $\lambda=-1$ must be an eigenvalue of $A$.
I have proven this in a long-winded way and I was wondering if these is a quick way of seeing it.
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Do all orthogonal matrices satisfy both $Q^TQ = I$ and $QQ^T = I$?
Some definitions of orthogonal matrix (for instance, on Wikipedia) use the definition:
$$
Q^TQ=QQ^T=I
$$
While other definitions (for instance, on Wolfram MathWorld) only include one of the products:
$$
QQ^T = I
$$
I remember reading that $Q^TQ$…
mai
- 197
1
vote
1 answer
Number of generators of a $2\times 2$ real orthonomal matrix inconsistent with visualization
Let $ A $ be a real orthonormal matrix. From the definition $ A^T A = I $, it is easy to see that $ \det(A) = \pm 1 $, hence a real orthonormal matrix should correspond to rotation and reflection.
If $ A $ is $ N\times N $, then number of generators…
Fanurs
- 63
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1 answer
Finding symmetric positive definite matrix A for A-conjugate vectors.
Given two distinct vectors $u$ and $v$ in $\mathbb{R}^n$ such that $u \neq \lambda v$ for any $\lambda \in \mathbb{R}$, find a symmetric positive definite matrix $A \in \mathbb{R}^{n\times n}$ such that $u^TAv=0$. The vectors $u$ and $v$ are thus…
user402940
- 381
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0 answers
I'm getting that orthogonal matrices don't (necessarily) have orthogonal rows/columns
As the title says, here's my "proof":
Let U be some orthogonal matrix:
Uᵀ = U⁻¹
∴ U Uᵀ = Uᵀ U = I
Considering the ijth element:
(U Uᵀ)ᵢⱼ = (Uᵀ U)ᵢⱼ = δᵢⱼ
∑UᵢₖUᵀₖⱼ = ∑UᵀᵢₖUₖⱼ = δᵢⱼ
∑UᵢₖUⱼₖ = ∑UₖᵢUₖⱼ = δᵢⱼ
The scalar product between two vectors is…
David
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Prove or disprove: Special linear group is subgroup of orthogonal group
Let $H$ be the special linear group. $H=\{M \in M_n(\mathbb{K}): det(M) = 1\}$. Is $H$ a subgroup of the orthogonal group?
Counterexample:
Let $A = \left(
\begin{array}{ccc}
1 & 1 \\
0 & 1 \\
\end{array}
\right)$
Then $det(A) = 1$. But $AA^T \neq…
karnan
- 175
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0
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Eigenvalue of Orthogonal Matrix and Transpose
Verify that if $P$ is an orthogonal matrix and $x = Py$ then $y^Ty = x^Tx$.
Let $A$ be a real symmetric $n × n$ matrix. Then we know that there exists a real orthogonal matrix $P$ such that $P^TAP$ is diagonal. By using the transformation $x = Py$,…
stedmoaoa
- 351
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1 answer
Find a unitary matrix $U$ such that $Ux = x$ and $Uy = -y$ for given unit perpendicular vectors $x$ and $y$ of $\mathbb{R}^{n}$
Find a unitary matrix $\bf{U}$ such that $\bf{U}$$x = x$ and $\bf{U}$$y = -y$ for given $\bf {unit}$ perpendicular (orthogonal) vectors x and y of $R^{n}$
If matrix is not found then please tell me why such a unitary matrix exists.
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votes
1 answer
find an orthogonal basis for the space of solutions of the following equations
Q.find an orthogonal basis for the space of solutions of the following equations
2x+y-z=0
y+z=0
How select solution? And how solve this question? ?
user375414
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Orthogonal arrays
This is from a note I found
I don't understand why t cannot equal to 3? If we choose first three columns, each row appears three times, for example, (0,0,0) appears three times in the subarray, which exactly matches the critera which is "any N × t…
whoisit
- 811
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Is a matrix that is orthogonally diagonalizable a projection matrix?
If a have a matrix say $A$ that is orthogonally diagonalizable (i.e. it can be written as $\lambda_1u_1u_1^T+ \lambda_2 u_2u_2^T+\dotsc \lambda_nu_nu_n^T$ , where the $u_i$ are the eigenvectors of the matrix $A$ and $\lambda_i$ are the…
john_w
- 580