Questions tagged [orthogonality]

This tag can be used to refer to the orthogonality of a set of vectors, a matrix or a linear transformation.

Two non-zero vectors $v$ and $w$ in an inner product space are orthogonal if $\langle v, w\rangle = 0$. Note that $v$ and $w$ are orthogonl if and only if the line spanned by $v$ and the line spanned by $w$ are perpendicular (the angle between them is $90^{\circ}$ or $\frac{\pi}{2}$ radians).

A set of non-zero vectors is called (pairwise) orthogonal if each pair of vectors in the set are orthogonal. Note that if a set of non-zero vectors is orthogonal, then it is linearly independent. If in addition to being orthogonal, all the vectors in the set have length one, it is called an orthonormal set.

A square matrix $A$ is called orthogonal if it is invertible and $A^T = A^{-1}$. A matrix $A$ is orthogonal if and only if the rows of $A$ are orthonormal and the columns of $A$ are orthonormal.

A linear transformation $T$ from an inner product space $V$ to itself is called orthogonal if $\langle T(v), T(w)\rangle = \langle v, w\rangle$ for all $v, w \in V$. Note that if $V$ is finite-dimensional, and we fix a basis for $V$, then $T(v) = Av$ for some square matrix $A$. In that case, $T$ is orthogonal if and only if $A$ is an orthogonal matrix.

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What does it mean when two functions are "orthogonal", why is it important?

I have often come across the concept of orthogonality and orthogonal functions e.g in fourier series the basis functions are cos and sine, and they are orthogonal. For vectors being orthogonal means that they are actually perpendicular such that…
quantum231
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Cross product in higher dimensions

Suppose we have a vector $(a,b)$ in $2$-space. Then the vector $(-b,a)$ is orthogonal to the one we started with. Furthermore, the function $$(a,b) \mapsto (-b,a)$$ is linear. Suppose instead we have two vectors $x$ and $y$ in $3$-space. Then the…
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Symmetric matrix is always diagonalizable?

I'm reading my linear algebra textbook and there are two sentences that make me confused. (1) Symmetric matrix $A$ can be factored into $A=Q\lambda Q^{T}$ where $Q$ is orthogonal matrix : Diagonalizable ($Q$ has eigenvectors of $A$ in its columns,…
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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…
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Orthogonality and linear independence

[Theorem] Let $V$ be an inner product space, and let $S$ be an orthogonal subset of $V$ consisting of nonzero vectors. Then $S$ is linearly independent. Also, orthogonal set and linearly independent set both generate the same subspace. (Is that…
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Why is orthogonal basis important?

Lets take the $\mathbb{R}^3$ space as example. Any point in the $\mathbb{R}^3$ space can be represented by 3 linearly independent vectors that need not be orthogonal to each other. What is that special quality of orthogonal basis (extending to…
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Name for matrices with orthogonal (not necessarily orthonormal) rows

Is there a name for a matrix whose rows (or columns) are non-zero orthogonal vectors ? It seems to me that "orthogonal matrix" would be a good name, but this is already taken -- it refers to a matrix whose rows (or columns) form an orthonormal set…
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Why is the derivative of a vector orthogonal to the vector itself?

$R(t) \cdot R'(t) = 0$, which is what every source I can find tells me. Even though I understand the proof I don't understand the underlying concept. If $R(t)\cdot R'(t) = 0$, then $R'(t)$ is orthogonal to $R(t)$, right? But you use the same…
Arvind Jeyabalan
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What can be said about a matrix which is both symmetric and orthogonal?

I tried to find matrices A, which are both orthogonal and symmetric, this means $A=A^{-1}=A^T$. I only found very special examples like I, -I or the matrix $$\begin{pmatrix} 0 &0& -1\\ 0& -1& 0\\ -1& 0& 0 \end{pmatrix} $$ Can a matrix…
Peter
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Is there an easy way to find the sign of the determinant of an orthogonal matrix?

I just learned that if a matrix is orthogonal, its determinant can only be valued 1 or -1. Now, if I were presented with a large matrix where it would take a lot of effort to calculate its determinant, but I know it's orthogonal, is there a simpler…
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What does it mean for two matrices to be orthogonal?

Firstly, please bear in mind that I do not have much knowledge about linear algebra. Secondly, I am not asking about some mathematical definitions but rather a more physical definition of this mathematical meaning. So, the problem is that I can…
TheQuantumMan
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Why is the matrix-defined Cross Product of two 3D vectors always orthogonal?

By matrix-defined, I mean $$\left\times\left = \left| \begin{array}{ccc} i & j & k\\ a & b & c\\ d & e & f \end{array} \right|$$ ...instead of the definition of the product of the magnitudes multiplied by the…
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What is the difference between orthogonal and orthonormal in terms of vectors and vector space?

I am beginner to linear algebra. I want to know detailed explanation of what is the difference between these two and geometrically how these two are interpreted?
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Orthogonal matrix norm

If $H$ is an orthogonal matrix, then $||H||=1$ and $||HA||=||A||, \forall A$-matrix (such that we can writ $H \cdot A$). What norm is this about?
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Double orthogonal complement is equal to topological closure

So I'm in an advanced Linear Algebra class and we just moved into Hilbert spaces and such, and I'm struggling with this question. Let $A$ be a nonempty subset of a Hilbert space $H$. Denote by $\operatorname{span}(A)$ the linear subspace of all…
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