0
votes
0answers
7 views

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 $w_i \neq z_j$ ...
2
votes
0answers
34 views

Proof of Gersgorin Discs Theorem

I have a question about the proof of Gersgorin Theorem from the book Matrix Analysis by Horn & Johnson. The Theorem states that for any $A\in \mathbb{C}^{n \times n}$ 1) all eigenvalues are ...
0
votes
1answer
69 views

Any $2\times 2$ complex matrix A is similar to one of these three: (See first line of the question)

(i) : $\left(\begin{array}{ll} \lambda_{1} & 0\\ 0 & \lambda_{2} \end{array}\right)$, (ii) : $\left(\begin{array}{ll} \lambda & 0\\ 0 & \lambda \end{array}\right)$, (iii) : ...
1
vote
2answers
65 views

What's the fastest way to determine Eigenvalues & Eigenvectors of any 2 by 2 Matrix?

My instructor claims that it's inefficient and superfluous to compute eigenvectors de novo for each $2$ by $2$ matrix. He suggested a trick instead which resembles the eigenvectors and cases here. ...
1
vote
1answer
32 views

Proof: If $F^3 = F$ then F is diagonalisable

let $V$ be a $\mathbb{R}$-vectorspace with $dim V < \infty$ and $F$ an endomorphism of V with $F^3 = F$. Show: F is diagonalisable. $F^3 = F$ is equivalent to $F^3 - F = 0$. Now I know that ...
3
votes
1answer
42 views

A Hermitian matrix $(\textbf{A}^\ast = \textbf{A})$ has only real eigenvalues - Proof Strategy [Lay P397 Thm 7.1.3c]

Would someone please explain the proof strategy at Need verification - Prove a Hermitian matrix $(\textbf{A}^\ast = \textbf{A})$ has only real eigenvalues? I brook the algebra so I'm not asking about ...
1
vote
1answer
34 views

Proof concerning eigen values

Could somebody help me into proving this theorem? if $A$ and $B^{H}$ are in $C^{m\times n}$ with $m\geq n$, then $\lambda (AB) = \lambda(BA) \cup \lbrace 0, \ldots ,0\rbrace.$ Thenks, Elnaz
3
votes
0answers
73 views

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 ...
6
votes
5answers
351 views

Proof Strategy - Prove that each eigenvalue of $A^{2}$ is real and is less than or equal to zero - 2011 8C

Remember that we've already proven the following, for any real symmetric $n\times n$ matrix $M$: (i) Each eigenvalue of $M$ is real. (ii) Each eigenvector can be chosen to be real. (iii) Eigenvectors ...
1
vote
2answers
64 views

Linear Algebra need help with proof please over eigenspaces

I know that if x and y are distinct eigenvalues of an nxn matrix A, then the intersection of eigenspaces is the 0 vector. How can I prove this?
5
votes
2answers
309 views

Proof Strategy - Tricky Question involving distinct, non-zero eigenvalues of $A^{2}$ - 2011 8C

Remember that we've already proven the following, for any real symmetric $n\times n$ matrix $M$: (i) Each eigenvalue of $M$ is real. (ii) Each eigenvector can be chosen to be real. (iii) Eigenvectors ...
-1
votes
4answers
87 views

Given the set of eigenvalues of a diagonalizable matrix, show that it satisfies an equation

Let $A$ be an $n \times n$ diagonalizable matrix. $A$ has only $2$ and $4$ as its eigenvalues. Show that $A^2 = 6A − 8I$. I get stuck on this question for a while. Can anyone give me a hint for this ...
0
votes
1answer
50 views

Prove that the eigenvectors are independent.

Given two vectors $\boldsymbol\alpha=\left(\alpha_1,...,\alpha_N\right)^{T}$ and $\boldsymbol\beta=\left(\beta_1,...,\beta_N\right)^{T}$, let $M$ be the $N\times N$ matrix whose entries are expressed ...
1
vote
2answers
57 views

need help on proof question on matrices MEI FP2

Matrix M is (n × n). For n=2 and n=3 prove that if the sum of the elements in each row of M is 1, then 1 is an eigenvalue of M. I know that to find eigenvalues and corresponding eigenvectors, the ...
3
votes
2answers
130 views

REVISTED$^1$: Circumstantial Proof: $P\implies Q \overset{?}{\implies} Q\implies P$

To prove that if a matrix $A\in M_{n\times n} ( F )$ has $n$ distinct eigenvalues, then $A$ is diagonalizable is enough to show that the opposite holds? That is, if $A$ is diagonalizable, then $A$ has ...
0
votes
1answer
95 views

Eigen value minimization-Proof

The two minimization problems below are equivalent: $\min\{\mathrm{trace}(AX^TBX): XX^T=I_n\}=\min\{\mathrm{trace}(AQ^T\tilde{B}Q): QQ^T=I_m\}$, where $A,\tilde{B}$ and $Q$ are square matrices of ...
1
vote
1answer
87 views

Is this a valid proof of eigenvectors/values of nonzeroilty?

Let $A$ be a square real matrix such that it is symmetric, show that if $Ax = \lambda x$ for some nonzero vector in $\mathbb{C}^n$, then in fact, $\operatorname{Re}(x)$ is an eigenvector of $A$ if it ...