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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63 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 ...
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2answers
20 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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1answer
21 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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1answer
29 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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0answers
15 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 ...
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0answers
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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1answer
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 ...
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1answer
12 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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1answer
15 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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0answers
18 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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1answer
26 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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2answers
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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0answers
29 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 $$w_1=\frac{v_1}{\|v_1\|}$$ ...
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4answers
46 views

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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2answers
32 views

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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0answers
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 ...
0
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2answers
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 $\{v_1, v_2\}$: $$ ...
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1answer
20 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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2answers
18 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*} ...
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1answer
34 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 ...
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2answers
68 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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2answers
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 ...
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2answers
18 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 ...
0
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1answer
9 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 ...
3
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0answers
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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0answers
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 ...
0
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1answer
25 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 ...
0
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1answer
43 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 ...
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2answers
55 views

Gram Schmidt and Inner Product

I'm working on this problem: Show that if $(e_1...e_n)$ is an orthonormal basis constructed using the Gram Schmidt process from $(v_1...v_n)$, then for any $j,$ $\langle e_j, v_j \rangle > 0$. ...
2
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0answers
29 views

Is this geometric Interpretation of $Q^T$ being orthonormal if $Q$ is orthonormal valid?

I was reading the book - Linear Algebra and its Applications, when I saw - Remark 2. Since $Q^T = Q^{-1}$, we also have $QQ^T = I$. When Q comes before $Q^T$, multiplication takes the inner ...
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0answers
42 views

Finding if the orthonormal basis of P1 with a given inner product

Hi, So I know for a basis to be orthonormal, the inner product needs to be 0 between components. So the standard basis <1,t> would not work in this case, as the integral from 1 to zero of t is ...
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0answers
40 views

orthonormal basis in $L^2$ space

Let $\{\phi_i (x)\}_{i=1}^\infty$ be an orthonormal basis for $L^2 (S)$. Prove that $\{\psi_{ij} (x,y) = \phi_i (x) \phi_j (y)\}_{i,j=1}^\infty $ is an orthonormal basis for $L^2 (S \times S)$. Thanks ...
0
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1answer
8 views

A list is a basis if norm of difference with orthonormal basis is bounded by given constant less than unity

This is a problem from Linear Algebra Done Right, 3rd Edition, problem 14 in 6.B. Given $\{e_1, \dotsc, e_n\}$ is an orthonormal basis of $V$, and $||v_j -e_j|| < \frac{1}{\sqrt{n}} \; \forall \; ...
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1answer
26 views

SVD of (2,1,-2) not ok

I'm trying to find the SVD of $$ \begin{pmatrix} 2&1&-2\\ \end{pmatrix} $$ I found $$\Sigma , u$$ But on the V matrix I got $$ \begin{pmatrix} ...
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3answers
46 views

Prove that $C^nx = \frac{1}{2^n}au + \frac{1}{3^n}bv$, for every $n \in N$

EDIT: I forgot to mention $C = 0.5uu^T + 0.33vv^T$ and now if I use it, I solve it easily. Given: $Cx = \frac{1}{2}au + \frac{1}{3}bv$, $x \in R^2$, $u,v$ are orthonormal vectors in $R^2$, $x = au ...
0
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2answers
27 views

How does one write an transformation matrix under an orthonormal basis?

I have an orthonormal basis (consisting of either vectors or polynomials). How do I write a matrix for a linear transformation Tv=w for any vector v and w in the vector space V? Is there a general ...
2
votes
2answers
35 views

Unitary Matrix columns = Orthonormal Basis

So I'm trying to understand why the columns of a unitary matrix form an orthonormal basis. I know it has something to do with the inner product, but I don't fully understand that either (we learned ...
0
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0answers
34 views

Projection of inner product spanned by orthonormal basis

So the question asks: Consider the inner product $⟨f,g⟩=$$ \begin{align} \int_{-1}^{1} f(x)g(x) \ dx &\end{align}$$ $ on $P_2$, the space of all polynomials of degree 2 or less. Find the ...
0
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1answer
40 views

Normalize vector : $ \left(\frac{e^{\frac{-i\pi}{4}}}{\sqrt{10}}, -3\frac{e^{\frac{i\pi}{4}}}{\sqrt{10}} \right)$

This is a homework question that I am struggling with. I have to normalize the vector $$ \left(\frac{e^{\frac{-i\pi}{4}}}{\sqrt{10}}, -3\frac{e^{\frac{i\pi}{4}}}{\sqrt{10}}\right)$$ as I have to ...
2
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0answers
21 views

Jordan Normal Form of a self-adjoint Linear Transformation

Let $V$ a finite inner product space, $dim V = n \geq 3$. Let $w_1,w_2 \in V$ such that: $<w_1,w_2>=0$, $||w_1||=||w_2||=1$ where $||w||$ is the norm of a vector $w$. The inner product ...
0
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0answers
21 views

Orthonormal basis and vectors in a space V

Let's say I have an orthonormal basis of $V$ $(e_1,\ldots,e_n)$ and a bunch of vectors in $V$ $(v_1,\ldots,v_n)$ that may or may not be linearly independent. How can I then prove that those vectors ...
0
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0answers
26 views

Showing the Haar wavelet is a complete and orthonormal sequence within $L_2[0,1]$

I define the mother Haar wavelet to be: \begin{align} \phi(t) = \begin{cases} 1 &\mbox{if } 0 \leq t < 1/2 \\ -1 & \mbox{if } 1/2 \leq t \leq 1 \\ 0 &\mbox{otherwise}. ...
0
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1answer
33 views

Finding the Orthonormal Basis of a given Subspace

Hi, So I am not sure how to do this problem. I think what I have to do is find a basis for the space of 2x2 matrices that are orthogonal to the identity matrix s.t. the HS inner product. And from ...
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0answers
24 views

The direction of a normal vector

This is from my textbook I'm a little bit confused with the direction arrow of the normal vector, shouldn't it be the green vector?
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0answers
24 views

QR Factorization of a rank 1 real matrix

An answer in my textbook says that QR Factorization is possible for any real matrix, but doesn't one need to have a full column rank in order to do this (through Gram-Schmidt process), as this is what ...
0
votes
1answer
27 views

How to find orthonormal bases of a null space?

$\mathbf{H}$ is a $K\times N$ matrix of complex elements where $N>K$. I guess there must be some standard algorithms to find a set of orthonormal basis for its Kernel, i.e. set of all ...
1
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1answer
59 views

Finding the orthogonal projection of a given vector on the given subspace W of the inner product space V.

Let $V = R^3$ with the standard inner product $u = (2,1,3)$ and $W = \{(x,y,z) : x + 3y - 2z = 0\}$ I came up with the basis $\{(-3,1,0), (2,0,1)\}$ but these are not orthogonal to each other. I'm ...
0
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0answers
67 views

matrix optimization in diagonal and orthonormality constraint

I'd like to solve the following optimization problem. Y is a CxN matrix A is a CxC and diagonal matrix Q is a CxC and orthonormal matrix X is a CxN matrix $$ min_{A,Q} {|| Y − AQX ||}^2_F $$ The ...
0
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0answers
22 views

Orthonormal Basis of Hermitian matrices for a Hilbert Space of Operators

Consider a set of operators $O$ on a Hilbert space $V$ of dimension $d$. I could prove that $O$ is also a Hilbert space with dimension $d^2$ (inner product being $(A,B) = tr(A^\dagger B))$. Now I am ...