# Questions tagged [orthogonality]

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

1,527 questions
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### Evaluation of generalized Laguerre function integrals using orthogonality relations

(NB - I am not asking to be spoon-fed with complete solutions, just pointing out any useful transformations, or giving general pointers would suffice.) The orthogonality relation for generalized ...
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### Bias-Variance OLS via eigendecomposition of projection matrix

I am struggeling to derive the squared-bias and variance based on an eigendecomposition for the OLS-procedure. The model Consider the univariate model $y_i = f(x_i) + \epsilon_i, \ i = 1, \dots, n$ ...
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### Write the set of vectors that are orthogonal to $v$ as a linear combination of two unit vectors.

$$v = \langle 1, -\sqrt 8, -\sqrt 8\rangle \text{ is a vector.}$$ I know I have to find two unit vectors $u$ and $w$ so that any vector that is orthogonal to $v$ can be expressed as a linear ...
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### What is the intuition behind $\langle I- \frac{xx^T}{\|x\|_2^2},\frac{xx^T}{\|x\|_2^2}\rangle=0$?

For any non zero vector $x$ the following inner product is zero meaning that these two matrices are orthogonal to each other. $$\langle I- \frac{xx^T}{\|x\|_2^2},\frac{xx^T}{\|x\|_2^2}\rangle=0$$ ...
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### Determine if a matrix is an orthogonal projection matrix

We define an orthogonal projection as a linear transformation that maps a vector into its orthogonal projection in some (given ahead) subspace $W$. Let's call the matrix of that transformation (...
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### Find the involutions in the indefinite orthogonal group O(2,1)

I would like to find the (linear) involutions that conserve the quadratic form $x^2+y^2-z^2$. Finding reasonable equations for the entries of the matrix is possible, but not particularly nice or ...
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### Find a set of orthogonal vectors in $\mathbb{R}^{n}$ dimension with their components summing to zero

I am interested in finding a set of, say 3, orthogonal vectors whose components add up to zero. This is a concept that is used in statistics as well, but for the sake of this question, I wanted to ...
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### Higher dimensional cross product equivalent

I'm working on a computer vision script for high dimensions that is highly reliant on the cross product in 3D, but as far as I know, it is only formally defined in 3D and 7D. However, experimentally, ...
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### determinant of two orthogonal matrices are zero when det(A)=det(B) [closed]

I am trying to solve following problem. Let $A,B \in \Bbb R^{n×n}$ be orthogonal matrices and $\det(𝐴)=\det(𝐵)$. . How can be proven that $𝐴+𝐵$ is singular? I know how to prove if the case ...
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### $\{(X,Y)\in \Bbb R^3: |X|=|Y|=1, \langle X,Y \rangle=0\}$ is homeomorphic to $\text{SO}(3, \Bbb R)$

Let $R=\{(X,Y)\in \Bbb R^3: |X|=|Y|=1, \langle X,Y \rangle=0\}$ with $\langle\,\cdot\,,\,\cdot\,\rangle$ the Euclidean scalar product. Prove that $R$ is homeomorphic to $\text{SO}(3, \Bbb R)$. I ...
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### An orthogonal projection induced by an $m \times n$-matrix

I am reading "Interlacing Eigenvalues and Graphs" by Willem H. Haemers. Right at the start in the proof of theorem 2.1 there is a step (marked in a red box below) which I do not understand. My ...
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### Is every orthogonal projection continuous?

Is there an example of a non-continuous linear operator $\pi$ on a $\mathbb R$-Hilbert space $H$ with $\pi^2=\pi$ and orthogonal null space and range? Clearly, if the range is closed, then $\pi$ is ...
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### Why can't the row space and nullspace be two lines in R3?

I'm learning Linear Algebra using Gilbert Strang's lectures and at lecture 14 he said the following: "Imagine two perpendicular lines in R3. Can they be the row space and the nullspace? No.". So the ...
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### Understanding a disprove example on Orthogonality subject

I was trying to disprove the following thoerem: If $A$ and $B$ are subspaces of $\mathbb{R}^n$ then $(A\cap B)^\bot = A^\bot \cup B^\bot$. I know that the theorem is not true, but my book gave the ...
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### Let $w_1,w_2,\dots,w_n$ be an orthonormal basis of $W$. If $v = a_1\cdot w_1+a_2\cdot w_2+\dots+a_n\cdot w_n$, then $a_1 = ?\ a_2=?\ a_n = ?$

Let $w_1, w_2,\dots, w_n$ be an orthonormal basis of $W$. If $v = a_1\cdot w_1+a_2\cdot w_2+\dots+a_n\cdot w_n$, then $a_1 = ?\ a_2=?\ a_n = ?$ How do I find the scalars $a_1, a_2$, and $a_n$ ?
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### Orthogonal projectors inequality

Let $V$ be a vector space and $V_1 \subset V_2$ two subspaces of $V$. Then, denoting by $P_1$ and $P_2$ the orthogonal projectors on $V_1$ and $V_2$ respectively, it holds for any matrix $A$ of ...
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### Gram--Schmidt ortogonalization for complex functions

I am trying to perform the Gram--Schmidt orthogonalization on a set of continuos complex functions. Numerical solution would be quite sufficient for me. I have three continuous functions of complex ...
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### Euclidean distance of set of orthogonal vectors

Let's define $x$ as a vector in $\mathbb R^n$ Let's define $V$ as the set of all vectors orthogonal to $x$, i.e $V$={$y$ in $\mathbb R^n$|$x·y=0$} Let's define $z$ as another vector in $\mathbb R^n$ ...
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### Give the conditions on the $m\times1$ vector so that a matrix $H$ is orthogonal

Give the conditions on the $m\times 1$ vector $x$ such that the matrix $H=I_m-2xx'$ is orthogonal. The only solution I have found and verified is $x$ could be the zero vector; however, I know there ...
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### Finding the orthogonal complement of a subspace of $\mathbb{R}^4$

Let $V$ be the vector space $\{(x,y,z,w)\in\mathbb{R}^4:x+y-z=0\ \text{and}\ x+y+w=0\}$. Then, what would be a basis for the orthogonal complement of $V$. I think we have to find the null space of ...
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### Why are $(\,f_1,…,f_n)$ linearly independent if $\|\,f_k-e_k\|_2<\dfrac{1}{\sqrt n}$, where $(e_k)$ is an orthonormal basis? [duplicate]

Let $(e_1,...,e_n)$ be an orthonomal basis and $(\,f_1,...,f_n)$ vectors such that $\|f_k-e_k\|_2<\dfrac{1}{\sqrt n}\,\forall k\,$. I'd like to show that $(\,f_1,...,f_n)$ are linearly independent....
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I'm trying to prepare for an exam and came across the following question: Given $n \times n$ matrix A, let U represent row space and W represent column space. A) Prove: $W \subseteq$ $U^{\... 0answers 32 views ### Gram-Schimdt get the Legendre polynomial When I apply the Gram-Schmidt algorithm I don't understand why I don't get the Legendre polynomials. When I apply this algorithm I always get monic polynomials whereas the Legendre polynomials aren't ... 2answers 29 views ### Proving$\dim(E_0) \geq n - k$I found a question from an old exam which I am not really able to wrap my head around. The questions states: Given$k<n$and$v_1, v_2, ..., v_k \in \mathbb{R} ^n$non-zero vectors, orthogonal to ... 2answers 52 views ### How do you find a point Q on the line L such that PQ is perpendicular to L P is the point (1,1,1) and the line L is given by the equation x ¯ = t ( 1 ... 1answer 42 views ### calculate orthogonal matrix Given matrix$A(m\times n)$find matrix$B(n\times m)$that fulfill the equation$A\,B=0\,(m\times m)$mean orthogonal m less then n 1answer 28 views ### Linear Algebra, orthogonal columns and length Suppose A is a 3x3 matrix whose columns are orthogonal and the length (two-norm) of each column equals 4. Then what is$A^T*A$? ... 1answer 19 views ### Question regarding orthogonality and linear independence I am new to linear algebra and was a bit confused regarding the following… Any feedback would be really appreciated... True or false? If$K$is a non-empty set of vectors in$R^n$, then$(K^\bot)^\...
I am new to linear algebra, as am having some doubts regarding the following question: True or False $u,v \in R^n$ $\left\lVert u\right\rVert=\left\lVert v\right\rVert$ if and only if u+v and u-v ...