# Tagged Questions

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### Convex combination of polynomials with roots on the unit circle and companion matrix

Given two $N^{th}$ order polynomials $P_0(z)$ and $P_1(z)$, let their roots be $w_k$ and $z_k$ respectively. All the roots of both polynomials lie on the unit circle $\mathcal{U}$. Also $w_i \neq z_j$ ...
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### Eigenvalue formula for 4x4 symmetric matrix

Is there a formula/algorithm that is accurate to used in finite precision arithmetic (aka numerical stable ) for small symmetric matrix of size 4x4. Additionally I'm looking if it require similar ...
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### Matrix representation of a linear operator

As I'm studying for my final, my book keeps skipping alot of steps and I don't know how tthey get from point a to point b - probably because its elementary at that stage in the book, except not to me ...
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### Criteria for a solvable septic equation

I have a certain 7x7 matrix whose elements are all symbolic, and I want to know the eigenvalues. I have to solve a septic equation, but it is generally impossible. However, I am only interested in the ...
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### Calculating the characteristic polynomial

I'm stuck with this problem, so I've got the following matrix: $$A = \begin{bmatrix} 4& 6 & 10\\ 3& 10 & 13\\ -2&-6 &-8 \end{bmatrix}$$ Which gives me the following ...
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### Proof for the form of characteristic polynomial

I'd like to proof: The caracteristic polynomial of $A \in M(n\times n, K)$ has the form: $P_A(\lambda) = (-1)^n \lambda^n + (-1)^{n-1} \operatorname{tr}(A)\lambda^{n-1} +\dots +\det(A)$ My proof ...
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### The minimal polynomial of A is dividing $x^{2013} -1$, prove A is diagonalizable over the complex field

$A$ is $nxn$ real matrix. The minimal polynomial of A is dividing $x^{2013} -1$. I need to prove that: (1). A is diagonalizable over the complex field. (2). If A is diagonalizable over the reals, ...
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### Eigenvalues of 3x3 Covariance Matrix, Geometric Interpretation

Problem Definition I would like to code an algorithm for decomposing a covariance matrix into its eigensolution (set of eigenvalues and corresponding eigenvectors. In my specific case I want to deal ...
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### Linear algebra : eigenvalues of an integral operator on polynomials

Consider the linear transformation $$T : \left\{ \begin{array}{ccc} \mathbb{R}_n[X] & \to & \mathbb{R}_n[X] \\ P & \mapsto & \int_0^1 (X + t)^n\,P(t)\,dt \end{array}\right.$$ where ...
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### Eigenvalue and Eigenvector proof of matrix polynomial

Consider a polynomial $$f(s) = s^m+a_1s^{m-1}+\cdots +a_m$$ and a square matrix $A$. Prove: if $\lambda$ is an eigenvalue of $A$ with corresponding eigenvector $x$, then $f(\lambda)$ is an eigenvalue ...
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### Question about Axler's proof that every linear operator has an eigenvalue

I am puzzled by Sheldon Axler's proof that every linear operator on a finite dimensional complex vector space has an eigenvalue (theorem 5.10 in "Linear Algebra Done Right"). In particular, it's his ...
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