# Tagged Questions

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### 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 ...
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### Adjacency matrix of directed graph

I am given adjacency matrix $A$ of directed graph. $A(x,y)$ counts the number of edges from $x$ to $y$. I want to show that if $A$ has constant outdegree $d$: (i) For any eigenvalue $\lambda$, we ...
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### Spectrum of product from two selfadjoint matricies

If I have 2 selfadjoint matricies A,B given, is the spectrum of $A B$ always real? I know that $A B$ is not necessary a selfadjoint matrix, but are some properties of the spectrum preserved?
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### 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 ...
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### 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 ...
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### 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 ...
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### 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) ...
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### real spectrum of an almost symmetric stochastic matrix

Let $M$ be a real nonnegative square matrix ''almost'' symmetric ($A_{ij}=0$ if and only if $A_{ji}=0$). In addition, $M$ is irreducible (not necessarily primitive) and row stochastic (the sum of each ...
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### Are any matrices positive semidefinite, non-negative, and not diagonally dominated?

If so, I'd appreciate any examples. Thanks.
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### Eigenvalues of $A^{T}A$

Let $\lambda_{i}(M)$ denote the $i$th eigenvalue of the square matrix $M$, and $T$ denote the matrix transpose. Is it true that $|\lambda_{i}(A^{T}A)|=|\lambda_{i}(A)|^{2}$ for every square matrix ...
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### $L^p$ norm of diagonal is $\leq$ Schatten $L^p$ norm

Let $A = (a_{ij})$ be an $n\times n$ symmetric matrix. Its Schatten norm is defined by $\|A\|_{S^p}^p = \sum_{j=1}^n |\lambda_j|^p$, where $\lambda_j$ are the eigenvalutes of $A$. I am trying to prove ...
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### Reconstructing a Matrix in $\Bbb{R}^3$ space with $3$ eigenvalues, from matrices in $\Bbb{R}^2$

I have a matrix which represents a closed loop matrix of a control system with delays (Control Systems Theory) in $\Bbb{R}^3$ space that has $3$ eigenvalues. Through some process I have obtained three ...
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### Decompose a complex symmetric matrix to retain positive definitness

I have a complex symmetric matrix $A$, (i.e. non-Hermitian and obeying $A=A^T$), which is positive definite, in the sense that: $$\Re({z^HAz}) > 0$$ for any $z$. I am able to verify this ...
Let $A$ be a $k\times k$ positive definite symmetric matrix. By spectral decomposition, we have $$A = \lambda_1e_1e_1'+ ... + \lambda_ke_ke_k'$$ and $$A^{-1} = ... 0answers 107 views ### Spectrum of Transition Matrix for Random Walk Consider the symmetric random walk on \{0, 1, \dots, n\} with transition probabilities P(j \to j \pm 1) = 1/2 for 1 \le j \le n-1 and P(0 \to 0) = P(0 \to 1) = P(n \to n) = P(n \to n-1) = 1/2. ... 1answer 206 views ### operator norm and spectral radius is it true that the operator norm of a matrix A is smaller than 1 if its spectral radius \rho(A) is smaller than 1? many thanks for any help, it is much appreciated! 1answer 347 views ### Spectral radius and positive definite of matrices Denote  \rho(A) to be the spectral radius of a matrix A, that is the maximal eigenvalue of A. We say that a matrix M is positive definite, respectively positive semidefinite, if x^TMx>0 ... 1answer 232 views ### Similar matrix proof A and B are similar matrices, if B=PAP^{-1} holds for a square, non-singular matrix P. Now am wondering if S^{-1}T and S^{-1/2}TS^{-1/2} are similar matrices? Am looking for a proof for it ... 0answers 66 views ### Root Convergence rate of Iterative Scheme [closed] I have an iterative sequence for optimizing an EM algorithm based loss function L(X) with t being the iteration number as: X_t=ABX_{t-1}+CX_{t-1}+X_{t-1} where A is a diagonal matrix, B and ... 1answer 929 views ### are there any bounds on the eigenvalues of products of positive-semidefinite matrices? I have real positive semidefinite matrices (symmetric) A and B, both are n \times n. I am looking for upper bounds and lower bounds on the m-th largest eigenvalue of AB, in terms of the ... 1answer 153 views ### The row- and column-sums of a nonengative matrix with spectral radius less than 1 Is it true that if a matrix has nonnegative elements and spectral radius less than 1, than the sum of its elements on each row (and column) is less than 1? Edit: What if the matrix has positive ... 1answer 187 views ### Behavior of the spectral radius of a convergent matrix when some of the elements of the matrix change sign I want to prove (or disprove) the following statement: If A is a square matrix with non-negative elements that has spectral radius less then 1, then any matrix obtained from A by ... 1answer 284 views ### How to solve X*A=C matrix equation where two (X and A) matrices are unknown? I have a spectroscopy problem that boils down to a matrix equation where X*A=C. I take N observations each consisting of 3 detector readings and my detectors suffer from some amount of cross-talk ... 3answers 994 views ### Ratio of largest eigenvalue to sum of eigenvalues — where to read about it? Let E_j be the jth largest-magnitude eigenvalue of a real symmetric N \times N matrix M. I've found that the ratio$$\frac{|E_1|}{\sum_{j=1}^N{|E_j|}}, is a measure of the "rank-one-ness" ...
From a bank of masters exams: Say the position of a particle moving in $\mathbb{R}^n$ is given by a smooth vector-valued function $\vec{x}(t)$. Suppose that $\vec{x}(t)$ satisfies a ...