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COMP3670: Introductiion to Machine Learning
创建日期：2024-04-18 16:13:37
标签： COMP3670
Introductiion to Machine Learning

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COMP3670: Introductiion to Machine Learning

Question 1 Properties of Eigenvalues (5+5=10 credits)
Let A be an invertible matrix.
1. Prove that all the eigenvalues of A are non-zero.
2. Prove that for any eigenvalue λ of A, λ−1 is an eigenvalue of A−1.
Question 2 Properties of Eigenvalues II (10 credits)
Let B be a square matrix. Let x be an eigenvector of B with eigenvalue λ. Prove that for all integers
n ≥ 1, x is an eigenvector of Bn with eigenvalue λn.
Question 3 Distinct eigenvalues and linear independence (20+5 credits)
Let A be a n× n matrix.
1. Suppose that A has n distinct eigenvalues λ1, . . . , λn, and corresponding non-zero eigenvectors
x1, . . . ,xn. Prove that {x1, . . . ,xn} is linearly independant.
Hint: You may use without proof the following property: If {y1, . . . ,ym} is linearly dependent
then there exists some p such that 1 ≤ p < m, yp+1 ∈ span{y1, . . . ,yp} and {y1, . . . ,yp} is
linearly independent.
2. Hence, or otherwise, prove that for any matrix B ∈ Rn×n, there can be at most n distinct
eigenvalues for B.
Question 4 Properties of Determinants (10+15=25 credits)
1. Prove det(AT ) = det(A).
2. Prove det(In) = 1 where In is the n× n identity matrix.
Question 5 Eigenvalues of symmetric matrices (15 credits)
1. Let A be a symmetric matrix. Let v1 be an eigenvector of A with eigenvalue λ1, and let v2 be an
eigenvector of A with eigenvalue λ2. Assume that λ1 6= λ2. Prove that v1 and v2 are orthogonal.
(Hint: Try proving λ1v
T
1 v2 = λ2v
T
1 v2. Recall the identity a
Tb = bTa.)
Question 6 Computations with Eigenvalues (3+3+3+3+3=15 credits)
Let A =
[−1 2
3 4
]
.
1. Compute the eigenvalues of A.
2. Find the eigenspace Eλ for each eigenvalue λ. Write your answer as the span of a collection of
vectors.
3. Verify the set of all eigenvectors of A spans R2.
4. Hence, find an invertable matrix P and a diagonal matrix D such that A = PDP−1.
5. Hence, find a formula for efficiently 1 calculating An for any integer n ≥ 0. Make your formula
as simple as possible.
1That is, a closed form formula for An as opposed to multiplying A by itself n times over.

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关键词一 关键词二 关键词三 关键词四 关键词五 和法规和的发更好的风格和 ECON 615 FINC 612 CSC148 COMP 639 ACCT 203 ACCT 211 PSYC 101 MTH2009 COMP52315 PSYCO 432 statistics MATH4063 11 xxx 1 MA1MSP MAST30028 MATH3888 CS341 EEE30002 ES2F5 MCD4160 CC0933 ECON 1101 MCHM3001 CHEM3118 COMMGMT 2511 COMP41280 STAT 631 24T1 IRE379 SENG365 ECON433 FIT3139 MATH4602 LING5621M FINS5516 ECON3208 MN2177 L2 SM EEET1415 Finance DS4023 COMP3004 7CCSMRTS K1S5B6 Stat230 CSCI4211 CSI2132 ECON30130 MAS6100 ENVS222 COM2107 INFO1112 STAT 4004 FIT2100 comp2022 COMP4691 Physics 307 Econ 659 probability CSCI316 fir29 CSCI203 COMP3160 COMP6714 INFO1113 Physics 7B Physics Photonics S203031 STAT2005 COMP90015 CSCI 2110 CSCC46 Information CSC477 COMP4650 Statistical COMP 346 COMP1017 ELEC340 MATH1007 Programming function STAT 244 CS 134 Java cse13s Math 108A FW20 ECON3389 CSE 130 Science 331 COMP9020 engineering COSC 445 CSE 5523 GAME2012 visualization MATH 235 Math 370 CSC343 COGS 108 EE524 COMP10200 STAT 416 3FK4 MSCI 311 ECON6113 ACC40700 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