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Papers (31)
2025
mscds 11
2024
mscds 10 pgmath 1 ugmath 20
2023
pgmath 3 ugmath 16
2022
pgmath 2 ugmath_22may 16 ugmath_23may 16
2021
pgmath 1 ugmath 15
2020
ugmath 15
2019
pgmath 2 ugmath 10
2018
pgmath 2 ugmath 14
2017
pgmath 1 ugmath 13
2016
pgmath 2 ugmath 11
2015
pgmath 1 ugmath 13
2014
ugmath 9
2013
pgmath 8 ugmath 10
2012
pgmath 2 ugmath 14
2011
pgmath 1 ugmath 12
2010
pgmath 4 ugmath 19
2023 pgmath

3 maths questions

Q5 Invariant lines and eigenvalues and vectors Eigenvalue constraints from matrix properties View
Consider the real matrix
$$A = \left( \begin{array} { l l } \lambda & 2 \\ 3 & 5 \end{array} \right)$$
Assume that $-1$ is an eigenvalue of $A$. Which of the following are true?
(A) The other eigenvalue is in $\mathbb { C } \backslash \mathbb { R }$.
(B) $A + I _ { 2 }$ is singular.
(C) $\lambda = 1$.
(D) Trace of $A$ is 5.
Q19 Groups Symplectic and Orthogonal Group Properties View
Let $U ( n )$ be the group of $n \times n$ unitary complex matrices. Let $P \subset U ( n )$ be the set of all finite order elements of $U ( n )$, that is, $P = \left\{ X \in U ( n ) \mid X ^ { m } = 1 \text{ for some } m \geq 1 \right\}$. Show that $P$ is dense in $U ( n )$.
Q20 Groups Algebraic Number Theory and Minimal Polynomials View
Let $A$ be a non-trivial subgroup of $\mathbb { R }$ generated by finitely many elements. Let $r$ be a real number such that $x \longrightarrow r x$ is an automorphism of $A$. Show that $r$ and $r ^ { - 1 }$ are zeros of monic polynomials with integer coefficients.