# linear algebra

1. ### I Meaning of each member being a unit vector

Summary: Meaning of each member being a unit vector, and how the products of each tensor can be averaged. Hello! I am struggling with understanding the meaning of "each member is a unit vector": I can see that N would represent the number of samples, and the pointy bracket represents an...
2. ### Decompose Involutory Matrix into Difference of Two Idempotents

I feel confused about proving the two terms are idempotents.
3. ### Simplifying a matrix algebra equation (revised)

I have a matrix equation (left side) that needs to be formatted into another form (right side). I've simplified the left side as much as I could but can't seem to get it to the match the right side. I am unsure if my matrix algebra skills are lacking or if I somehow messed up the starting...
4. ### Basis of eigenvectors

Okay so I found the eigenvalues to be $\lambda = 0,-1,2$ with corresponding eigenvectors $v = \begin{pmatrix} 1 \\ 1 \\ 1 \end{pmatrix}, \begin{pmatrix} 1 \\ 0 \\ 1 \end{pmatrix}, \begin{pmatrix} 1 \\ 1 \\ 0 \end{pmatrix}$. Not sure what to do next. Thanks!!!
5. ### Python Invert a matrix from a 4D array : equivalence or difference with indexes

I have a 4D array of dimension $100\text{x}100\text{x}3\text{x}3$. I am working with `Python Numpy. This 4D array is used since I want to manipulate 2D array of dimensions $100\text{x}100$ for the following equation (it allows to compute the $(i,j)$ element $F_{ij}$ of Fisher matrix) ...
6. ### Expectation value of operators and squeezing in the even cat state

I started and successfully showed that the expectation of X_1 and X_2 are zero. However the expectation value of X1^2 and X2^2 which I am getting is <X1^2> = 0.25 + \alpha^2 and <X2^2> = 0.25. How do I derive the given equations?
7. ### I Question about an eigenvalue problem: range space

A theorem from Axler's Linear Algebra Done Right says that if 𝑇 is a linear operator on a complex finite dimensional vector space 𝑉, then there exists a basis 𝐵 for 𝑉 such that the matrix of 𝑇 with respect to the basis 𝐵 is upper triangular. In the proof, he defines U=range(T-𝜆I) (as we have...
8. ### Proof a property for a 3x3 matrix

Let a 3 × 3 matrix A be such that for any vector of a column v ∈ R3 the vectors Av and v are orthogonal. Prove that At + A = 0, where At is the transposed matrix.
9. ### I Linear Algebra - Inner Product problem

I need help to know if I'm on the right track: Prove/Disprove the following: Let u ∈ V . If (u, v) = 0 for every v ∈ V such that v ≠ u, then u = 0. (V is a vector-space) I think I need to disprove by using v = 0, however I'm not sure.
10. ### I Linear least squares regression for model matrix identification

Summary: I need to Identify my linear model matrix using least squares . The aim is to approach an overdetermined system Matrix [A] by knowing pairs of [x] and [y] input data in the complex space. I need to do a linear model identification using least squared method. My model to identify is a...
11. ### Vector space has dimension less than d

1. Homework Statement Problem given to me for an assignment in a math course. Haven't learnt about roots of unity at all though. Finding this problem super tricky any help would be appreciated. Screenshot of problem below. 2. Homework Equations Unsure of relevant equations 3. The...
12. ### Isomorphisms preserve linear independence

1. Homework Statement Let $T:V \rightarrow W$ be an ismorphism. Let $\{v_1, ..., v_k\}$ be a subset of V. Prove that $\{v_1, ..., v_k\}$ is a linearly independent set if and only if $\{T(v_1), ... , T(v_2)\}$ is a linearly independent set. 2. Homework Equations 3. The Attempt at a...

28. ### I HHL Algorithm for Solving Linear Equations

I have a question about HHL algorithm https://arxiv.org/pdf/0811.3171.pdf for solving linear equations of the form: A x = b Where A, x and b are matrices Take for example 4x1 + 2x2 =14 5x1 + 3x2 = 19 HHL apply the momentum operator eiAτto/T on the state, do a Fourier Transform on |b> and...
29. ### Courses On the benefits of retaking advanced linear algebra

I'm a physics student who has the option to take some advanced math courses (Real analysis through Rudin and beyond, functional analysis if I have time, as well as algebra through Artin). I'm only just going into my second year this term, and will either be retaking linear algebra 2, or taking...
30. ### I Confused by this result for the tensor product of two vectors

Given two probability distributions $p \in R^{m}_{+}$ and $q \in R^{n}_{+}$ (the "+" subscript simply indicates non-negative elements), this paper (page 4) writes down the tensor product as p \otimes q := \begin{pmatrix} p(1)q(1) \\ p(1)q(2) \\ \vdots \\ p(1)q(n) \\ \vdots \\...