# MATH 33AH Lecture Notes - Lecture 22: Symplectic Matrix, Symmetric Matrix, Block MatrixExam

by OC2716258

Department

MathematicsCourse Code

MATH 33AHProfessor

AllStudy Guide

FinalThis

**preview**shows half of the first page. to view the full**1 pages of the document.**Math 33A Practice Problems

November 30, 2016

Here you’ll ﬁnd a few practice problems for the exam. It’s important to understand

that these problems were not vetted by Dr. Shlyakhtenko, and should not be

considered an indication of the problems you’ll see on the exam.

1. Let J0be the following 2n×2nblock matrix:

J0=0−In

In0.

A 2n×2nmatrix Ψ is called symplectic if ΨTJ0Ψ = J0.

(a) Show that all symplectic matrices have determinant ±1.

(b) Show that if Ψ and Φ are symplectic matrices, then so are Ψ + Φ, ΨΦ, Ψ−1, and

ΨT.

(c) Characterize all 2 ×2 symplectic matrices.

(d) Consider the map ω:R2n×R2n→Rdeﬁned by

ω(~u, ~v) = −~uTJ0~v.

Show that the 2n×2nmatrix Ψ is symplectic if and only if

ω(Ψ~u, Ψ~v) = ω(~u, ~v)

for all vectors ~u, ~v ∈R2n.

2. Suppose Ais a 2n×2northogonal matrix, and that det A= 1. Show that the algebraic

multiplicity of 1 as an eigenvalue of Ais even.

3. Show that if Ais odd-dimensional and skew-symmetric (i.e., AT=−A), then Ais not

invertible.

4. Show that if ~v1and ~v2are eigenvectors of the symmetric matrix Awith distinct eigen-

values λ1and λ2, then ~v1and ~v2are orthogonal.

5. Let bbe some ﬁxed value and consider the quadratic form

q(~x) = kx2

1+bx1x2+kx2

2.

Characterize the deﬁniteness of qin terms of k. That is, determine the values of k

for which qis positive- (or negative-) deﬁnite, positive (or negative) semi-deﬁnite, or

indeﬁnite.

1

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