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So I am trying to find the spectrum of square matrix A where A=[ 1 1...... 1 1 1 .... ... ... ... ....... ... 1 ]. I basically am trying to put a sparse matrix, not sure how to post it here. but basically the whole square matrix has 1s everywhere. So, I am not sure how to find the spectrum of A. I am not even sure what Spectrum means. I do now that matrix A is linearly dependent, and this the determinant would have to equal 0. When I computed the eigenvalues of A, I get 0 and 1. So, my question is what is Spectrum and how would you find the spectrum of A.

Sarah Smith
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  • I didn't mean sparse.the matrix does not have any zero entries. The whole square matrix has 1s. the dimension of square matrix is not specified. it is an n x n matrix. So, can you please help me find a duplicate or help answer my question? Thanks – Sarah Smith Feb 01 '20 at 02:42
  • Seems to me that you ought to start by reviewing your course materials so that you know what “spectrum” means. It’s going to be really hard to compute it without that. – amd Feb 01 '20 at 02:55
  • I mean I do have linear algebra knowledge like with linearly independence/dependence, determinants, and I know what when a matrix is invertible, nullspace,etc. This is more like upper division course that has material that builds on to linear algebra. What material are you inferring? – Sarah Smith Feb 01 '20 at 03:02
  • The adjective you are looking for is low-rank rather than sparse. The singular value decomposition (SVD) of this matrix has only one term ${\tt 11}^T$ where ${\tt 1}$ denotes the all-ones vector. So it's actually a rank-one matrix, which comes with some interesting properties. – greg Feb 01 '20 at 03:09

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The all ones square matrix can be written as $A=ee^T$ where $e= (1,1,...,1)^T$.

It is easy to see that $A$ is symmetric real so all eigenvalues are real.

If $v \bot e$ we see that $Av = 0$ and $Ae = ne$ so $A$ has $n-1$ zero eigenvalues and one positive eigenvalue of $n$.

copper.hat
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  • The most novel aspect of this Question is asking for the definition of spectrum of a matrix. The spectrum is a fancy name for the collection of a matrix's eigenvalues (roots of its characteristic polynomial). Since eigenvalues can be repeated, the spectrum is a natural instance of a multiset, where items can be present in a collection with repetitions (unlike the members of a set, which are always distinct). The number of repetitions of an eigenvalue is its (algebraic) multiplicity. – hardmath Feb 01 '20 at 16:25