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In Arveson's book A Short Course on Spectral Theory, on page 64 (section on spectral measures) the author mentions the usual spectral decomposition of a normal operator $N$ as $$N=\sum_{\lambda \in ... 0answers 24 views ### Define a positive dot product in \mathbb{R^3} Consider the matrix A= \begin{bmatrix} k & k-1 & 0 \\ 1-k & 2-k & 0 \\ 2k-3 & 2k-1 & 2 \end{bmatrix}  with k \in \mathbb{R} and let be f_a: \mathbb{R^3} \rightarrow ... 1answer 55 views ### A characterization of a certain family of matrices in terms of another matrix. Consider a real matrix A of dimension n \times n. Assume k \leq n is given. I am looking for ways to describe the following set of matrices in terms of properties of A. \mathcal{S}(A) = \{B ... 1answer 67 views ### Continuity of the spectral radius Let M \in \mathbb{R}^{n\times n} be a nonnegative irreducible matrix with strictly positive entries on its main diagonal. Then M is primitive and by the Perron-Frobenius Theorem we know that the ... 1answer 67 views ### Finding the smallest max eigenvalues for related matrices? While messing around with a spectral approach to a graph coloring question, I happened upon a type of problem I hadn't seen before. Suppose you have two symmetric n x n matrices in the form ... 1answer 18 views ### Normal Matrices Unitarily Diagonazible Disclaimer: This thread is just meant to record (Q&A). Are the unitarily diagonazible matrices precisely the normal ones? Surely, every normal matrix has an eigenbasis. Now I got asked by a ... 1answer 24 views ### Non-Unitarily Diagonalizable Matrices When searching for matrices that are similar to a diagonal matrix but not in a unitary way then a first hint would be to exclude the normal ones. But apart from that is there a general form for such ... 1answer 40 views ### Prove that spec(f(A)) = f(spec(A)). Can someone please explain this proof to me? Thanks! Let A \in \mathbb{C}^n and let f(x) be a polynomial. Prove that spec(f(A)) = f(spec(A)) (where if S \subseteq \mathbb{C}, f(S) := \{ ... 1answer 14 views ### Consider a symmetric matrix X with eigendecomposition X=UVU^T, how to call \sum_{v_{k,k}>0}v_{k,k}u_ku_k^T? Consider a symmetric matrix X with eigendecomposition X=UVU^T How do people call \sum_{v_{k,k}>0}v_{k,k}u_ku_k^T? Sum of positive components of X? The positive semi definite part of X? ... 2answers 36 views ### Example: Algebraic Multiplicity vs Geometric Multiplicity Is there a simple example of a matrix having an eigenvalue whose geometric multiplicity is strictly smaller than its algebraic multiplicity? 0answers 48 views ### Infinite Dimensional Vector Space Equivalence of Positive Matrix What is the corresponding linear operator on an infinite dimensional vector space, say a Banach space or Hilbert space, to the nonnegative matrix on a finite dimensional vector space? What is the ... 1answer 26 views ### logarithm of projection I want to prove what's used in the fourth line below the "Proof" section here: http://en.wikipedia.org/wiki/Quantum_relative_entropy#The_result The statement is: Let \rho be a density operator on a ... 1answer 30 views ### Lower bound for the spectralradius of a matrix Any submultiplicative norm (for example the row-sum-norm) is an upper bound for the spectralradius of a matrix A. But is there a way to get a suitable LOWER bound for the spectralradius ? ... 1answer 79 views ### Eigenvalues of polynomials of a matrix and its inverse up to summation by identity There is a paper that I am reading and the following has been considered without proof: (Suppose \lambda(.) defines the spectrum of a matrix and one can define a random variable on this spectrum say ... 1answer 75 views ### Left and right eigenvectors perpendicular to each other I just read in a textbook on numerical methods that you can always have that the right eigenvectors of a matrix can be taken as orthonormal to the left eigenvectors for a diagonalisable matrix. This ... 1answer 47 views ### Generalized eigenvalue problem; why do real eigenvalues exist? Under this text, you can see two pairs of matrices (A,C). I am currently solving the generalized eigenvalue (A-\lambda C)v=0 for several pairs of (A,C) and found out that they also have real ... 0answers 32 views ### Eigenvalues of the linear operator T^*T Let T be a linear operator T: H_1 \mapsto H_2, where H_1 and H_2 are both Hilbert Spaces. Suppose further that T is bounded, but not self adjoint. Suppose I also know that for functions ... 1answer 118 views ### Condition for degenerate eigenvalues for a matrix Given a diagonalizable matrix M (that is, a normal matrix), can we determine whether the matrix has degenerate eigenvalues without explicitly calculating all the eigenvalues and eigenvectors? 1) An ... 2answers 35 views ### Spectral Radius and Norm of multiplied vector Let \mathbf{A}, \mathbf{B} be square matrices of equal dimensions, \mathbf{w} a vector of compatible dimensions and \rho be the spectral radius operator. Does the following hold? If \rho ... 2answers 40 views ### the spectrum of a matrix If A is an n \times n nilpotent matrix show that I-A is invertible then find the spectrum of I-A ? for part one i've shown that I-A is invertible by finding its inverse using that A is a ... 1answer 86 views ### Spectral decomposition of normal operator Define T from L_{2}(R) into itself by T(f)(t)=f(t+1). Show that T is normal and finds its spectral decomposition. I've shown that f is normal (in fact it's unitary) but how do I find its ... 0answers 100 views ### Spectral decomposition of TT^* On l_{2} let T be given by Te_{n}=\frac{e_{n+1}}{n+1} where (e_{n})_{n\ge1} is the canonical orthonormal basis. Find the spectral decomposition of TT^*. I find that ... 1answer 57 views ### Spectrum of idempotent element Let A be some unitary algebra over \mathbb{C}. If a^2=a and 0\ne a\ne 1 then \{0,1\}\subset \sigma_A(a) (\sigma_A(a) is the spectrum if a). I believe that also \sigma_A(a)\subset ... 0answers 13 views ### Any way to simplify this expression? So I have a vector of asset allocation weights given by x \in R^4 and a covariance matrix of the asset returns \Sigma \in R^{4,4}. I know by the spectral theorem, \Sigma = V DV^{-1} and the ... 2answers 45 views ### Distorted Unitary matrices Let U be an unitary and D be a diagonal matrix. We know that for all vectors v on the sphere Uv is on the sphere and,$$\langle Uv,Uv\rangle=\langle v,v\rangle.$$What are the vectors v on ... 0answers 50 views ### Vectors on a Sphere Let S be a sphere centered at origin in \Bbb R^{2n} of radius \sqrt{2n}. Let D be a diagonal matrix. Let U be unitary matrix. Let r\in\Bbb Z_+ be a fixed integer. (1) For a vector v ... 3answers 135 views ### Spectrum of a compact operator If the spectrum of a compact operator is finite, I don't understand why 0 has to be a member. I have proved that for all \epsilon > 0, there is only a finite number of eigenvectors which have ... 1answer 48 views ### Perturbation of a matrix with negative eigenvalues Let A be a square matrix with all eigenvalues negative. What is the relationship between the \lambda_\max of perturbed matrix A + X and the norm of the perturbation \|X\|? PS: I know that the ... 1answer 59 views ### Fast Gauss-Seidel convergence on low rank matrices I stumbled upon the following remarkable fact when experimenting with the Gauss-Seidel iterative solver: First I construct a low-rank symmetric positive semi-definite matrix A = M^TM with M a ... 0answers 50 views ### Characterization of all matrices with unit spectral radius under constraint Let A \in \mathbb{R}^{n \times n}_{\geq 0} be a symmetric matrix with positive row sums \mathbf{d} := A\mathbf{1} > 0. I am interested in characterizing all those positive diagonal matrices Z ... 1answer 51 views ### relations between two linear operators Let \alpha,\beta be linear operators on a finite dimensional vector space V over field F. Let \gamma=\alpha\circ\beta and \delta=\beta\circ\alpha. Prove that: (1). m_\delta(x) divides ... 0answers 35 views ### Eigenvalues and eigenvectors of AB and BA in Endomorphisms [duplicate] Let A and B endomorphisms of a finite vector space over a field F , Prove or a counterexample to the following statements: a) all eigenvector of AB is an eigenvector of BA; b) all eigenvalue of AB is ... 1answer 77 views ### Prove that if T a normal linear transformation and invertible, then T^{-1} is normal. The question is: Prove that if T a normal linear transformation and invertible, then T^{-1} is normal. Then I have to find the spectral decomposition of T^{-1}. At first I tried to prove it by ... 0answers 234 views ### Singular value decomposition of an arbitrary anti-symmetric (A=-A^{T}) complex matrix I am a physicist and very much used to the fact that any self-adjoint matrix (H^{\dagger} =H) in a finite-dimensional complex linear space can be uniquely specified by (a) the set of its (real) ... 1answer 34 views ### Does non-Hermitian implies at least one complex eigenvalue? Ok so I'm studying linear algebra and we went trough the Spectral Theorem, including and proving the fact that for every Herimitian matrix, its eigenvalues have 0 imaginary parts. I was wondering is ... 1answer 366 views ### Why does spectral norm equal the largest singular value? This may be a trivial question yet I was unable to find an answer:$$\left \| A \right \| _2=\sqrt{\lambda_{\text{max}}(A^{^*}A)}=\sigma_{\text{max}}(A)$$where the spectral norm \left \| A \right ... 1answer 614 views ### The eigenvectors of a matrix and its transpose that correspond to the same eigenvalue are not orthogonal Spent hours trying to prove this after encountering it in Lax's discussion of the spectral theorem, but no luck. Here's the problem (it is Theorem 18 in Lax 2ed, Chapter 6): A mapping A has ... 1answer 67 views ### Spectrum of Symmetrizable Matrix A matrix  M  is symmetrizable if  M = D S  with  D  a square diagonal matrix with positive entries, and  S  a symmetric matrix. What can be said about the spectrum of  M ? It seems like I ... 1answer 34 views ### “Almost orthogonalizing” matrices using a signature matrix Suppose A and B are two real symmetric n \times n matrices (If simpler, consider A and B to be 0/1 matrices, say, adjacency matrices of d-regular graphs). Then ||AB||_{op} \leq ... 1answer 112 views ### expectation of norm of orthogonal projector The question has to do with calculating the expected squared norm of a random projection. We have a 2D subspace T := span\{U1, U2\} where U1 is a random vector uniformly distributed over unit ... 0answers 67 views ### Change of basis and spectral theorem I've been having trouble with such a rudimentary problem. Let us define a matrix A:$$A = \begin{pmatrix} 3 & 0 & -i \\ 0 & 3 & 0 \\ i & 0 & 3 \end{pmatrix}$$A is a 3 by 3 ... 0answers 91 views ### Examples of deeper results in finite-dimensional vector spaces? this one is a bit inverted! So I am busy doing an advanced undergrad course in Linear algebra, and it is going very well, the problems in the book seem fairly routine. To be able to see if I am any ... 1answer 135 views ### Prove a bilinear operator is symmetric and positive definite I'm having problem showing the following: All operators are defined on V which is real (not complex). Let f be a bilinear operator that is anti-symetric (meaning f(a,b)=-f(b,a)) and let J be ... 0answers 363 views ### Spectral Theorem for Commuting Self-Adjoint Operators Welcome everybody :) I'm working together with a little group preparing for the upcoming exams in "Mathematical Methods of Physics." There's one tricky task involving some linear algebra, namely the ... 1answer 253 views ### Affine transform of multivariate gaussian If X_1, \ldots, X_n are iid N(0,1) or in other words \mathbf{X}=(X_1, \ldots, X_n) is distributed N(\mathbf{0}, \mathbf{I}), then A\mathbf{X}+\mu is distributed N(\mu, AA^t). Showing that ... 1answer 260 views ### uu^T is the standard matrix for the orthogonal projection of \mathbb{R}^n on the subspace spanned by u Let A be an n\times n symmetric matrix, and u an eigenvector of A. Why is it true that \forall x\in\mathbb{R}^n, x^{T}uu^{T}(I-uu^T)x=0? If I'm able to show this is true then I can show ... 0answers 209 views ### Proof of the spectral theorem for normal operators from two lemmas I have the following lemmas that I can prove: Let T be a linear operator on a Hermitian space V and let W be a T-invariant subspace of V . Then W^⊥ is T^*-invariant Let T be a normal ... 0answers 36 views ### Inverting a discrete linear transformation Consider the transformation from the set \{a_n\}_{n=0}^N to the set \{p_j\}_{j=0}^N:$$ p_j = \sum_{n = 0}^Na_n\phi_n(x_j) where $\{\phi_n(x)\}_{n=0}^N$ is a set of basis functions (linearly ...
For a non-singular matrix, its pretty straightforward to prove that $\lambda$ is eigenvalue of $A$ if and only if $\frac{1}{\lambda}$ is eigenvalue of $A^{-1}$. Let $A$ be a non-singular matrix, $x$ ...
Why do I have: $Tr(SDS^{-1})=Tr(D)$?