For questions related to orthonormality. A set of vectors in an inner product space is called orthonormal if each vector is a unit vector, and vectors are pairwise orthogonal.

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Given A $m\times n$, $\{u_1,…,u_n\}$ ON basis of $R^n$, prove eigenvalues aren't negative [duplicate]

Given $A$ $(m\times n)$, $\{u_1,...,u_n\}$ ON(orthonormal) basis of $R^n$ which are eigenvectors of $A^TA$ with $\lambda_1 , ... , \lambda_n$ eigenvalues accordingly. Prove: Eigenvalues are not ...
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51 views

Does a orthogonal basis for the span of $S$ always have the same dimension as the basis of $S$

Does a orthogonal basis for the span of $S$ always have the same dimension as the basis of $S$ Basically if I have found the orthonormal basis for the span of S can I use that to find the dimension ...
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1answer
37 views

If $Tv=\mu v$ for some $\mu>0$, then $v\in\ker(T^{1/2})^\perp$

Let $V$ be a separable $\mathbb R$-Hilbert space $T$ be a bounded, linear, nonnegative and symmetric operator on $V$ $(v_n)_{n\in\mathbb N}$ be an orthonormal basis of $V$ with $$Tv_n=\mu_nv_n\;\;\;\...
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33 views

If $ι:U→V$ is a Hilbert-Schmidt embedding and $(v_n)_{n∈ℕ}$ is an orthonormal basis of $V$, then $(ιι^*v_n)_{n∈ℕ}$ is an orthonormal basis of $ιU$

Let $(U,\langle\;\cdot\;,\;\cdot\;\rangle_U)$ and $(V,\langle\;\cdot\;,\;\cdot\;\rangle_V)$ be separable $\mathbb R$-Hilbert spaces $\iota:U\to V$ be a Hilbert-Schmidt embedding $T:=\iota\iota^\ast$ ...
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56 views

For an orthogonal matrix $Q$, why does $QQ^T = I$?

In my linear algebra text (Strang), an orthogonal matrix is defined to be a square matrix whose columns are orthonormal. In other words, an orthogonal matrix is a matrix $Q = [q_1 \cdots q_n]$ where ...
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1answer
38 views

Find SVD of $A$

How do I find the singular values? They somehow show that $\lambda_1 = 27, \lambda_2 = 6, \lambda_3 = 0$. I still can't see how they found them with the equations I made in my solution.
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21 views

Orthonormal set of basis functions in $L^2([a,b])$

I am wondering about the basis functions for $$e_n(x)=\frac{e^{2\pi i n x/L}}{\sqrt(L)}$$ where $L = b - a$ on the domain of $L^2([a,b])$ when doing fourier series. Basically, we must scale it by $L = ...
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46 views

the proof of variational principal for the principal eigenvalue (checking orthonormal subset)

Hi I am looking at part 3 of the proof in Evans Chapter 6. I have difficulty understanding "Furthermore from (6) and (7) we see that $(\lambda_k^{-1/2} w_k)$ is an orthonormal subset of $H_0^1(U)$. ...
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1answer
80 views

Orthonormal basis with specific norm

I need to show that $v_1,...,v_n$ is basis for $V$ whenever $e_1,...,e_n$ is an orthonormal basis for V and $v_1,...,v_n$ are vectors in $V$ such that $$\left\Vert e_i-v_i\right\Vert < \frac{1}{\...
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1answer
53 views

How can we compute the square root of an operator of the form $Cv=\sum_{n\in\mathbb N}\langle v,e_n\rangle_Ve_n$?

Let $\mathbb K\in\left\{\mathbb C,\mathbb R\right\}$ $U$ and $V$ be $\mathbb K$-Hilbert spaces such that $U\subseteq V$ and that the inclusion $\iota$ is Hilbert-Schmidt $C:=\iota^\ast$ denote the ...
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2answers
39 views

Endomorphism is normal and idempotent iff it is an orthogonal projection.

I've searched for answers for this question here for some time but haven't found an applicable answer because I could only find related questions, but not this one in particular. Suppose $V$ is a ...
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3answers
50 views

Rule for $\langle x,y\rangle$ if we know orthonormal base?

How to define $\langle x,y \rangle$ in space of polinoms, where $1, x-1 , 1-x^2$ are orthonormal base($\Vert a\Vert = 1$, $\langle a1, a2\rangle = 0$)? I'm a bit lost, I know how to do it with my ...
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1answer
30 views

Riesz Representation Thereom for Polynomials with real coefficients problem

Find a polynomial q(x) $\in$ P$_2$($\Bbb R$) Such that $ p ( 1/4 ) = $$\int_0^1 p(x)q(x) \,dx$$ $. I'm sorry to ask this question, but I've been working on it for some time. The inner product on P$_2$(...
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36 views

Let $H$ be a Hilbert space, $V≤H$ be closed, $Q:H→V$ be the orthogonal projection, $(e_n)_{n∈ℕ}$ be an ONB of $H$. Is $(Qe_n)_{n∈ℕ}$ an ONB of $V$?

Let $\mathbb K\in\left\{\mathbb C,\mathbb R\right\}$ $U$ and $H$ be $\mathbb K$-Hilbert spaces $(e_n)_{n\in\mathbb N}$ be an orthonormal basis of $H$ $\iota:U\to H$ be an embedding and $V:=\iota(U)$ ...
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66 views

Let the column vectors of a $3 × 3$ matrix $A$ form an orthonormal basis. Explain why $A^T = A^{−1}$ .

Let the column vectors of a $3 × 3$ matrix $A$ form an orthonormal basis. Explain why $A^T = A^{−1}$ . My Attempt: $AA^T=I$ if and only if $A^TA=I$. So $A$ is orthogonal if and only if $A^T$ is ...
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25 views

Find the orthonormal projection?

I am trying to find the orthonormal projectiont of $X\in V\equiv M_{2\times2}(\mathbb{R})$ over the space of diagonal matrix defined by $$D=\{M=(m_{ij})\in V:m_{ij}=0,\ i\not=j\}\ \ \ i,j\in\{1,2\}$$ ...
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26 views

Transformation matrix is jordan normal form

I have the following question: Given a finite-dimensional, unitary vector space V and a endomorphism f on V, is it possible to choose an orthonormal basis B of V in such a way, that the transformation ...
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18 views

If $U_0,V$ are Hilbert spaces, $(e_n)$ is an ONB of $U_0$ and $ι:U_0→V$ is an embedding, can we complete $(ιe_n)$ to an ONB of $V$?

Let $U$ and $H$ be Hilbert spaces $Q\in\mathfrak L(U)$ be nonnegative and symmetric $U_0:=Q^{1/2}U$ be equipped with $$\langle u,v\rangle_{U_0}:=\langle Q^{-1/2}u,Q^{-1/2}v\rangle_U\;\;\;\text{for }...
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Derive the relations between cartesian co-cordinate systems.

Let $K$ and $\bar K$ be two cartesian co-orditate systems in $\mathbb{R}^3$. The element: $$s^2=(\Delta x^1)^2+(\Delta x^2)^2+(\Delta x^3)^2$$ is an invariant in all co-ordinate system. I want prove ...
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29 views

Find an orthogonal basis for $\mathbb P_2$

The problem: For polynomials $\mathbb{P_2}$ we define the inner product between p and q as: $$ \langle p,q\rangle =p(t_0)q(t_0)+p(t_1)q(t_1)+p(t_2)q(t_2) $$ with $$t_0=0, t_1=1, \textrm{ and } ...
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23 views

Finding orthogonal operators in $\mathbb{R}^2$

Let $V = \mathbb{R}^2$ where $V$ is an inner product space with dot product. Let $v \in V$ be a unit vector. I want to show there are exactly two orthogonal operators $T: V \to V$ such that $T((1,0)) ...
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70 views

Eigenvalues and eigenvectors of the Householder matrix $H = I - \frac{2}{u^Tu} uu^T$

So during my first revision for the semester exams, I went through exercises in books/internet and I found 2-3 that caught my eye. One of them was the following: Let $u \in \mathbb R^n$ be a non-...
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21 views

Orthonormal basis and matrix.

Let $E = \{e_1,_{...},e_n\} , F = \{f_1,_{...},f_n\} $ be 2 orthonormal basis of V. Does $[Id]_F^E $ the transfer matrix from E to F is orthogonal ? I think that the answer is yes, thought of shoving ...
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22 views

The signature of an inner product space does not depend on its basis

In R.W.R. Darling's "Differential Forms and Connections" an inner product is defined for a vector space $V$ as a bilinear, symmetric and nondegenerate (but not necessarily positive-definite) map from $...
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36 views

weak solution via Galerkin approximation

I am reading Evan's approach to Existence of weak solution via Galerkin method in some PDE notes. I have difficulty understand the following assumption i.e. Let "$\{w_k\}_{k=1}^\infty$ be an ...
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16 views

for every linear map $\ T:V\to V$ : $\ [T^*]_B=(M^t)^{-1}A^tM\ $ when $\ [T]_B=A$.

Let $V$ be an inner product space of finite dimension over $\mathbb{R}$ and Let $B=\{v_1,...v_n\}$ be a basis of V (not necessarily orthonormal). Let $M\in M_n(\mathbb{R})$ a matrix whose i,j element ...
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14 views

Extending $\{u_1, u_2\}$ to an orthonormal basis when finding an SVD

I've been working through my linear algebra textbook, and when finding an SVD there's just one thing I don't understand. For example, finding an SVD for the 3x2 matrix A. I will skip the steps of ...
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1answer
23 views

Finding an orthonormal basis for subspace intersection, $W^{\perp}\cap V$

I am stuck trying to find an orthonormal basis for $W^{\perp}\cap V$. I'm given V = span$\{v_1,v_2,v_3\}$ and that $$ v_1= \begin{bmatrix} 1 \\ 1 \\ 0 \\ 1 ...
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50 views

Generalized Hermite Function as eigenfunction of a differential operator

I'm going through this paper. The article defines function function $\phi_n^\mu(x)$ that is orthonormal on $L^2$ with measure $dm = dx$: \begin{equation} \phi^\mu_n =\left(\frac{\gamma_\mu(n)}{\...
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15 views

Find orthonormal bases for the orthogonal subset

Let S be the subset $$\{\left(1,0,2\right),\left(3,2,1\right),\left(1,-2,7\right)\}\subset\mathbb{R}^{3}$$ Find orthonormal bases for $S^{\perp}$ and $S^{\perp\perp}$ I have begun by putting these ...
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16 views

square eigenvectors for Singular Value Decomposition?

This is from my textbook What I don't understand is, $V$ and $U$ are already square, why the textbook says "if we want to make them square"?
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20 views

How to find a specific orthogonal basis for a hermitian Matrix

I got to show that every hermitian form has an orthognal basis. More precisely, that there exists a basis, so that the matrix of the hermitian form looks like $$ \begin{pmatrix} E_p & & \\ ...
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27 views

Rank of product of SVD vectors

Given a matrix $M$, we can compute its singular value decomposition $M=U\Sigma V^*$ where $^*$ is the complex conjugate transpose. $U$ and $V$ are unitary, so $UU^*=I$, $VV^*=I$. Let's take the $i$-th ...
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32 views

Transformation of Orthonormal Bases

Suppose that $u_1, . . . , u_n$ and $v_1, . . . , v_n$ are orthonormal bases for $\Bbb{R}^n$. Construct the matrix A that transforms each $u_i$ into $v_i$ to give $Av_1 = u_1, . . . Av_n = u_n$.
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33 views

Gram-Schmidt Process, finding orthonormal basis

Suppose I'm given $2$ random vectors $(v_1,v_2)$ I want to find orthonormal basis $(w_1,w_2)$ Are the following equivalent? for the $w_2$ case $$w_1=\frac{v_1}{\|v_1\|}$$ $$x_2=v_2-\frac{(v_1,v_2)}{...
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How do I find orthonormal basis of $U$?

Let $U$ be the subspace of $\mathbb{R}^5$, which is through $(1,2,3,-1,2)^T$ and $(1,0,-1,0,1)^T$ spanned. How do I find orthonormal basis of $U$?
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Let $u_1,u_2,\dots,u_n$ be a basis for $\mathbb{C}^n$ show that it is an orthonormal basis for some inner product.

Let $u_1,u_2,\dots,u_n$ be a basis for $\mathbb{C}^n$ show that it is an orthonormal basis with respect to some inner product. How would i go about doing this?
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17 views

Point in $U$ closest to $(2,0,1,6)$

Let $U$ be the intersection of $x_1+x_2+x_3+x_4=0$ and $x_1+2x_2+3x_3+4x_4=0$ in $\mathbb{R}^4$. The basis of $U$ is $((1,-2,1,0), (2,-3,0,1))$ and the basis of the orthogonal space $U^\perp$ is $((3,...
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41 views

Write a vector as a linear combination of orthonormal set vectors

How can I determine whether a vector can be expressed as a linear combination of a orthonormal set vectors ? Let's say I have a orthonormal set of vectors $\{v_1, v_2\}$: $$ v_1=\left(-\frac{1}{2},-\...
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1answer
22 views

Orthonormal Basis and Orthonormal Set

I was pretty confused about these two terms: 'orthonormal basis' and 'orthonormal set'. I know the orthonormal basis is just the normalized vectors of the orthogonal basis, but what about the ...
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19 views

summing inner product of orthonomal basis

I need some help with some very basic linear algebra when doing calculations in inner product space. Here is a line I got lost when reading a book... \begin{align*} (x,e_m)=\left(\sum_{n=1}^k\...
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1answer
35 views

Eigenvalues and Eigenvectors relating to orthogonal basis and diagonal matrices

Find the eigenvalues and eigenvectors of the matrix. $$A = \begin{bmatrix} 1 & 1 & 0 \\ 1 & 0 & -1\\ 0 & -1 & 1 \end{bmatrix}$$ As we have seen in the lectures,...
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70 views

Orthonormal Basis Question: Linear Algebra

I've been staring at this proof for a long time so any suggestions would be of great help! Prove that for any $m\times n$ matrix $A$ there is an orthonormal Basis $B =\{ v_1,\ldots,v_n\}$ of $\mathbb ...
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46 views

Proof of claim on orthonormal elements in an inner product space

Let $X$ be an inner product space and $\{e_{n}\}_{n=1}^{\infty} \subset X$ be an orthonormal set. Show that $$ \sum_{n=1}^{\infty}|\langle x,e_{n}\rangle\langle y, e_{n}\rangle| \leqslant \|x\|\|y\| ...
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24 views

Proving a basis for inner product space V when $||e_j-v_j||< \frac{1}{\sqrt{n}}$.

Suppose $(e_1,e_2,...,e_n)$ is an orthonormal basis of the inner product space $V$ and $v_1,v_2,...,v_n$ are vectors of $V$ such that $$||e_j-v_j||< \frac{1}{\sqrt{n}}$$ for each $j \in \left\{1,2,....
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1answer
10 views

Lower bound of coefficients in an orthogonal decomposition

Is there a theorem of this form? Given a vector $x$ and an ONB $\{v_1, v_2, \ldots, v_n\}$, $\exists \, c>0$ such that $\langle x, v_j \rangle \ge c ||x||$ for some $j$. In other words, when you ...
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26 views

Describing the Unit Normal to a Cylinder in $\mathbb{R}^{3}$

This is problem 34.4 in Munkres' Analysis on Manifolds. Let $\mathcal{C} = \{ \ (x,y,z) \in \mathbb{R}^{3}\mid x^{2} + y^{2} = 1 \text{ and } 0 \leq z \leq 1 \ \}$. Orient $\mathcal{C}$ by declaring ...
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14 views

additional strucutre

Do I use the wrong calculations to find orthogonal projection?? We need to find a Projection of X onto V. In my calculations I am getting the same result as in the textbook but multiplied by factor of ...
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1answer
28 views

Show that $A$ is orthogonal iff the column vectors of $A$ form an orthonormal basis

My book says that: (a) $A$ is orthogonal if and only if the row vectors of $A$ form an orthonormal basis of $\mathbb{R}^n$ under the euclidean inner product; and (b) $A$ is orthogonal if and only ...
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1answer
44 views

Finding an orthonormal basis with only one vector

I'm programming a little thing to project things to my screen from a variable amount dimensions, and when I tried to implement the ability to move the origin and rotate the direction looked in strange ...