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A2 Midterm Solution
Creation date：2024-05-21 17:54:50
Belonging category：数学mathematics
Tag A2
Midterm Solution

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Midterm Solution

Problem 1
1. True. Let Ax = λx with λ ∈ C and non-zero x ∈ Cn. Because A2 = A, Ax = A2x = Aλx =
λ2x = λx, which implies that (λ2 − λ)x = 0. Since x contains at least one non-zero element,
λ2 − λ = 0 must hold. It follows that λ = 0 or 1.
2. True. Note that the rank of a zero matrix is 0 and its dimension of kernel (null) space is n by the
rank-nullity theorem. Besides, Ker(A) = Ker(ATA) (Since if Ax = 0, ATAx = 0. Conversely,
if ATAx = 0, Ax = 0 follows from xTATAx = 0). Therefore, by the rank-nullity theorem,
rankA = n − dimKer(A) = n − dimKer(ATA) = 0, which implies that A is zero matrix. (An
alternative proof is that tr(ATA) = 0 implies
m∑
i=1
n∑
j=1
a2ij = 0. Thus aij = 0 where aij denotes the
element in the ith row and jth column of A)
3. True. Since A−B is real symmetric, there exist an orthogonal matrixO and a real diagonal matrix
D with diagonal elements {d1, d2, ..., dn} such that A − B = OTDO. From xT (A − B)x = 0,
xTOTDOx = 0. We can construct a series of vectors xi such that Oxi = ei where ei is the
standard unit vector (only i th element is 1 and other elements are 0 in this vector). Since
di = e
T
i Dei = 0 for i = 1, 2, ..., n, we conclude that D = 0 and it follows that A − B = 0 and
A = B.
Problem 2
1. Let A have distinct eigenvalues λ1, ..., λn with corresponding eigenvectors x1, ..., xn. If follows
that Axi = λixi for i = 1, ...n. Since A commutes with B, ABxi = BAxi = Bλixi = λiBxi
which implies that Bxi is also an eigenvector of A with respect to eigenvalue λi (when Bxi = 0,
the following analysis still holds). Therefore, Bxi lies in the eigenspace spanned by xi so that
there exists µi such that Bxi = µixi. Hence (µi, xi) is an eigenvalue and eigenvector pair for B,
for each i = 1, ...n. Let O = [x1, x2, ..., xn] and D be a diagonal matrix with diagonal elements
{µ1, µ2, ..., µn}, then BO = OD and B = ODO−1. Therefore, B is diagonalizable.
2. Let Λ be a diagonal matrix with diagonal elements λ1, ..., λn, then A = OΛO
−1. Since B =
ODO−1, it is sufficient to prove that there exist coefficients a0, a1, ..., an−1 such that D =
an−1Λn−1+ an−2Λn−2+ ...+ a1Λ+ a0I. Since D and Λ are diagonal matrices, µi = an−1λn−1i +
an−2λn−2i + ...+ a1λi+ a0 should hold for i = 1, ..., n. Equivalently, there should exist a solution
for the linear equation 
1 λ1 · · · λn−11
1 λ2 · · · λn−12
...
...
. . .
...
1 λn · · · λn−1n


a0
a1
...
an−1
 =

µ1
µ2
...
µn

1
where the leftmost square matrix denoted as V is a Vandermonde matrix and
det(V ) =
∏
1≤i(λj − λi)
is non-zero since A has distinct eigenvalues. Thus V is non-singular and there exists a unique
solution for the linear equation, which completes the proof.
Problem 3
1. Characteristic polynomial of A: A− λI = −λ3 + 2λ2 − λ = −λ(λ− 1)2.
Therefore, eigenvalues of A: 0 or 1.
2. Algebraic multiplicity for λ = 0 is 1, and for λ = 1 is 2.
3. Geometric multiplicity for λ = 0 is 1, and for λ = 1 is 1.
4. A matrix is diagonalizable if and only if the algebraic multiplicity equals the geometric multi-
plicity of each eigenvalue. Therefore, matrix A is not diagonalizable
5. For Av1 = 0, v1 =
 0−1
2
, For (A− I)v2 = 0, v2 =
 1−1
5
, For (A− I)v3 = v2, v3 =
 03
−5

6. P =
 0 1 0−1 −1 3
2 5 −5
 and rank of matrix P is 3. J = P−1AP =
0 0 00 1 1
0 0 1
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SVM. (10 points) Consider an SVM classifier using RBF kernel, and finish the following python code to search for the optimum kernel parameter γ. You are required to calculate the RBF function and search yourself, instead of using functions in Scikit-Learn (sklearn). def parameter search(gamma list ,Xtr,ytr,Xts,yts ,...): """ Parameters gamma list: A list of RBF kernel parameter gamma to search for Xtr: Training data, shape (ntr,nrow,ncol) ytr: Training labels, shape (ntr,) Xts: Test data, shape (nts,nrow,ncol) yts: Test labels, shape (nts,) ... Add other parameters as needed Returns: gamma optimum: The gamma with minimum error on Xts, yts """ ... return gamma optimum
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