1
vote
1answer
27 views

Relationship between matrix 2-norm and orthogonal basis of eigenvectors

Given the following matrix: $$ A = \left( \begin{array}{cc} 3 & 4 \\ 0 & 5 \\ \end{array} \right)$$ calculate $\|A\|_2$, with $\|A\|_2 = max_{x \in \mathbb{R}^2 -\{0\}} \frac{\langle Ax,Ax ...
2
votes
1answer
58 views

Show a linear transform is self adjoint - check my answer

We are given $T:V \to V$ a normal linear transform (meaning $TT^*=T^*T$) We are also given $T^2=T$. Show that $T$ is self adjoint (meaning $T^*=T$). What I did I think I may have done something ...
2
votes
1answer
235 views

prove that if $\lambda$ is an eigenvalue of T then $\bar\lambda$ is eigenvalue of $T^*$

I have to prove that if $\lambda$ is an eigenvalue of T then $\bar\lambda$ is eigenvalue of $T^*$ (adjoint) I know that $<Tv,u> = <\lambda v,u> = ...
1
vote
1answer
60 views

If $D$ is the operator of differentiation, prove $D^{2}$ is a self adjoint linear operator on V and find all its eigenvalues and eigenvectors

Suppose $V$ is the space of infinitely differentiable complex valued functions $f$ on $[0,\pi]$ such that $D^{2k+1}f(0) = 0 = D^{2k+1}f(\pi)$ for all integers $ k \geq 0$. Then V is a complex IPS with ...
0
votes
2answers
186 views

Unitary matrix with all its eigenvalues equal to 1 must be the identity matrix

Let $A$ be a unitary matrix over field $F$ ($F=\mathbb{R},\mathbb{C}$) Prove that if all its eigenvalues equal to $1$ then $A$ must be the identity matrix $I$. I am having a hard time figuring out ...
3
votes
2answers
1k views

Linear Algebra: Distance between two parallel lines

Find the distance between the two (obviously parallel) lines below where $\alpha ,\beta \in \mathbb R$ are scalars. $$\text{Line ...
1
vote
1answer
28 views

Problem from Roman: a lower bound for trace of |T|^2

I'm working on a problem from Stephen Roman's Linear Algebra text, #20 on p. 235: Suppose $\tau \in \mathcal{L}(\mathbb{C}^n)$ and let the characteristic polynomial $\chi_{\tau}(x)$ have roots ...
2
votes
2answers
99 views

Onto and one-to-one

Let $T$ be a linear operator on a finite dimensional inner product space $V$. If $T$ has an eigenvector, then so does $T^*$. Proof. Suppose that $v$ is an eigenvector of $T$ with corresponding ...
1
vote
1answer
42 views

How to prove that the corresponding matrix is unitary

Let's say we are given hermitian matrix $H$. How to prove that the matrix $M$, formed from eigenvectors of $H$ is unitary? Thanks
2
votes
3answers
87 views

Linear Algebra Question on Eigenvalues

I am having a difficult time with the following question. Any help will be much appreciated. Let $A$ be an $n×n$ real matrix such that $A^T = A$. We call such matrices “symmetric.” Prove that the ...
0
votes
1answer
237 views

Inner product on the vector space $p_2$ of polynomials of degree 2 or less

For this question, dene an inner product on the vector space of $P_2$ of polynomials, through the formula $$p(x)q(x) = \int p(x)q(x)dx$$ What are the lengths $$\|\ 1\ \|, \|\ x\ \|,\|\ x^2\ \|$$ ...
3
votes
1answer
289 views

Eigenvalues of symmetric matrix in real inner product space

I got the following exercise to solve: Let $A\in\mathbb{R}^{n\times n}$ be a symmetric matrix and let $\lambda_{1}\geq\lambda_{2}\geq\cdots\geq\lambda_{n}$ be its eigenvalues sorted in a ...
4
votes
1answer
97 views

inequality on inner product

Let $x \in \Bbb R^n$ and $Q \in M_{n \times n}(\Bbb R)$, where $Q$ is hermitian and negative definite. Let $(\cdot,\cdot)$ be the usual euclidian inner product. I need to prove the following ...
3
votes
1answer
559 views

Hermitian positive semi-definite-Square root

Problem: Let $A$ be a Hermitian positive semi-definite $n$ by $n$ matrix (The field of scalars is $\mathbb{C}$). Let $B$ be an $n$ by $n$ matrix that commutes with A. Prove that $B$ and $\sqrt{A}$ ...