MATH 342 Definitions

Question 1

Similar Matrices

Answer 1

Let A,Β ∈R^(nxn). Then A and B are similar if there exists an inverible matrix P ∈R^(nxn) s.t
A = PBP^-1

Question 2

Orthogonal Matrix

Answer 2

A square matrix A is an orthogonal matrix if:
A^TA = I
(A^T = A^-1)
(A has orthonormal columns)

Question 3

Diagonalizable matrix

Answer 3

An nxn matrix A is a diagonal matrix if there exists an nxn invertible matrix P and an nxn diagonal matrix D s.t.
A = PDP^1

Question 4

QR-Factorization

Answer 4

Let A be an mxn matrix with linearly independent columns. Then there exists a mxn matrix Q with orthonormal columns and an nxn upper triangular matrix R with positive diagonal entries s.t.
A = QR

Question 5

Singular Value Decomposition

Answer 5

Let A ∈R^(mxn) with rank(A) = r. Then there exists orthogonal matrices U ∈R^(mxm) and V ∈R^(nxn), and Σ ∈R^(mxn) with the form: [D 0, 0 0] with D = [σ1 … σr] ∈R^(rxr) a diagonal matrix and σ1 ≥ σ2 ≥…≥ σr >0 s.t.
A = UΣV^T

Question 6

Least-squares solution

Answer 6

Let A ∈R^(mxn) and b ∈R^m. A least squares solution to Ax = b is a vector x^ s.t
||b -Ax^|| ≤ ||b - Ax|| for all x∈R^n

Question 7

Spectral decomposition of a symmetric matrix

Answer 7

Let A ∈R^(nxn) be a symmetric matrix. Let u1, …, un be an orthonormal eigenbasis of A corresponding to eigenvalues of λ1, λ2, …, λn, respectively. Then
A = λ1u1u1^T + λ2u2u2^T + … + λnunun^T

Question 8

Positive definite matrices

Answer 8

Let A ∈R^(nxn) be a symmetric matrix. Then A is called positive definite if for any x =/= 0:
x^TAx > 0

Question 9

Cholesky Factorization

Answer 9

Let a be an nxn positive definite symmetric matrix. Then there exists an nxn upper triangular matrix R with positive diagonal entries s.t.
A = R^TR

Question 10

Rank Nullity Thm for an mxn matrix

Answer 10

Let A ∈R^(mxn). Then
Rank(A) + Nullity(A) = n

Question 11

Basis Thm for a vector space

Answer 11

Let V be a vector space with dimV = n. Let S ⊆V be a subset with exactly n vector. Then S is a basis of V if one of the following conditions holds:
1) S is linearly independent
2) SpanS = V

Question 12

Let A be an nxn matrix. State two conditions that are equivalent to “A is diagonalizable”

Answer 12

Let λ1, λ2,…, λk be all distinct eigenvalues of A. let Eλi be the eigenspace of λi.

1) dim(Eλ1) +…+ dim(Eλk) = n
2) let mi be the multiplicity of λi, and m1 +…+mk = n and dim(Eλi) = mi
for all i = 1, …, k

Question 13

Spectral Thm for symmetric matrices

Answer 13

A square matrix is symmetric if and only if it is orthogonally diagonalizable

Question 14

State an equivalent condition for a symmetric matrix being negative definite.

Answer 14

An nxn symmetric matrix A is negative definite if A has all negative eigenvalues.