Summary Linear Models in Statistics

LM Summary

Carine Wildeboer

January 2022

, Week 1
Lecture 1
1) A is symmetric ⇐⇒ A = A′ .
2) For conformable matrices, (ABC)′ = C ′ B ′ A′ .
A and B are conformable if their orders are s.t. AB is defined.
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3) Let A be a n × m matrix. Then: AA′ = j=1 aj a′j .
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4) If A, B and D have m columns and D is diagonal then ADB ′ = j=1 dj aj b′j , where aj and bj
are column vectors and the dj are scalars, j = 1, ..., m.
5) tr(A) = tr(A′ )
6) tr(AB) = tr(BA)
7) Let x,y and A be conformable, s.t. x′ Ay is a scalar. Then: x′ Ay = tr(x′ Ay) = tr(Ayx′ )
8) rank(A) = rank(A′ )
9) rank(AB) ≤ rank(A)
10) If A is n × m and B is a square matrix of rank m, then: rank(AB)=rank(A)
11) For any matrix A: rank(A)=rank(A′ A)=rank(AA′ )
12) The determinant of a matrix is nonzero ⇐⇒ the matrix has full rank.
13) If λ is a scalar and A n × n, then: |λA| = |λIn A| = |λIn ||A| = λn |A|
14) If A,B and C are non-singular matrices, then: (ABC)−1 = C −1 B −1 A−1
A matrix is non-singular if it is a square matrix with full rank.
15) If A is non-singular, then (A′ )−1 = (A−1 )′
16) The spectral decomposition or eigenvalue-eigenvector decomposition is given by:
S = P DP ′ , where:
- P is a square orthonormal matrix, with the eigenvectors of S as columns.
- D is a diagonal matrix with eigenvalues or latent roots or characteristic roots as
diagonal elements
17) If S is symmetric then |S| us given by the products of its eigenvalues.
18) If S is symmetric, then tr(S) is given by the sum of its eigenvalues.

Lecture 2
19) Is S is non-singular and S ≥ O, then S > O.
20) S ≥ O ⇐⇒ S = AA′ for some A.
21) S > O ⇐⇒ S = CC ′ for a non-singular C.
22) S ≥ O ⇒ A′ SA ≥ O.


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