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Linear Models > Linear Algebra Primatives > Flashcards

Linear Algebra Primatives Flashcards

1
Q

If matrix A has an inverse, then

A
  • the inverse is unique
  • A-1A=I
  • AA-1=I
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2
Q

(A-1)-1=

A

A

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3
Q

if A and B are non-singular matricies, then (inverse relations)

A

(AB)-1=B-1A-1

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4
Q

If A is non-singular and k ne 0, then (inverses)

A

(kA)-1=(1/k)A-1

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5
Q

If A and B are matrices such that AB is defined (ie conformable), then (transposes)

A

(AB)T=BTAT

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6
Q

A is called symmetric if

A

AT=A

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7
Q

If A and B are nxn square matrices, then (determinants)

A

det(AB)=det(A)det(B)

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8
Q

If A in nxn, then det(A) = 0 IFF

A

A is singular

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9
Q

The rank of A is

A

the greatest number of linearly independent columns (or rows) of A

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10
Q

If A and B are non-singular, then for any matrix C, (ranks)

A

C, AC, CB, ACB all have the same rank (assuming multiplications defined)

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11
Q

If A in an mxn matrix of rank r, then there exists non-singular matrices P and Q such that PAQ equals one of the following:

A
  • I (this is “eye”)
  • [I 0]
  • [I 0]T
  • [I 0 0 0] (square)
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12
Q

The rank of AB can not exceed

A

the rank of either the rank of A or B

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13
Q

If A is a nxn matrix, then det(A)=0 IFF (rank)

A

the rank of A is less than n

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14
Q

The matrix of a quadratic form can always be chosen to be

A

symmetric (attach proof)

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15
Q

A and B are said to be congruent matrices IFF

A

there exists a non-singular matrix, C, such that B=CTAC.

We say C is the congruent transformation of A.

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16
Q

Let A be an nxn symmetric of rank r. There exists a non-singular matrix C such that

A

CTAC = D where

D is a diagonal matrix with exactly r non-zero diagonal elements

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17
Q

If A and B are congruent matricies, then

A

they have the same rank

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18
Q

Let C be an mxn matrix with rank r, then the ranks of CTC and CCT are

A

also r.

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19
Q

Let A be an nxn matrix. There always exist n complex numbers …

A

eigenvalues (characteristic roots, symbol lambda) that satisfy

det(A-lambda I) = 0. If A is real symmetric, then all lambda’s are real.

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20
Q

Let A be an nxn symmetric matrix, the rank of A (in terms of eigenvalues)

A

is the number of non-zero eigenvalues

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21
Q

Let A be an nxn matrix. A has ____ eigenvalues IFF A is singular.

A

A has at least one zero eigenvalue.

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22
Q

Let A be an nxn matrix. The determinant of A (in terms of eigenvalues)

A

the product of its eigenvalues

Π lambdai

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23
Q

Let A be an nxn matrix, and let C be any nxn non-singular matrix. The following matrices all have the same eigenvalues

A
  • A
  • C-1AC
  • CAC-1
24
Q

Let A be an nxn real matrix. A necessary and sufficient condition that there exists a nonzero y that satisfies:

Ay=lambda * y

A

is that lambda be an eigenvalue of A.

25
Q

Let P be an nxn matrix. P is called orthonormal IFF

A

P-1=P’

THUS

P’P=I

26
Q

P’P=I suggests

A

P is orthonormal

27
Q

Let A be an nxn matrix, and let P be an orthonormal matrix, then (determinants)

A

det(A) = det(P’AP)

28
Q

Let x and y be nx1 vectors. x and y are called orthogonal if

A

x’y = 0

29
Q

Let A be an mxn matrix and B be a nxp matrix. And and B are said to be orthoganal to each other IF

A

AB=0

30
Q

Let A ben an nxn symmetric matrix. There exists an orthonormal matrix P such that

A

P’AP = D where Di is a diagonal matrix whose diagonal elements are the eigenvalues of A.

31
Q

Full rank factorization. Let A be an mxn matrix with rank r>0. There exists matrices AL (mxr with rank r) and AR (rxn with rank r) such that:

A

A=ALAR

32
Q

The column space of a matrix A is :

A

the set of vectors that can be generated as linear combinations of columns of A. (ditto for rowspace).

33
Q

Let A be an nxn matrix. A is called positive semidefinite if:

A
  • A=A’ (symtric)
  • y’Ay ge 0 for all y
  • There exists at least one y ne 0 such that y’Ay=0.
34
Q

If A is positive semidefinite, then

A
  • Rank of A is less than n
  • The eigenvalues of A are greater than or equal to 0
  • Let P be a nxn non-singular matrix. P’AP is also p.s.d.
35
Q

Let A be an nxn matrix. A is called positive definite if

A
  • A=A’
  • y’Ay>0 for all y ne 0
36
Q

If A is positive definite, then

A
  • The rank of A is n
  • All the eigenvalues of A are greater than 0
  • Let P be a nxn non-singular matrix. P’AP is also positive definite
37
Q

A matrix is called non-negative definite if

A

it is either positive definite or positive semi-definite

38
Q

Let C be an mxn matrix with rank r. C’C and CC’ are both

A

non-negative definite. They are positive definite IFF they have full rank.

39
Q

Let A be a nxn symmetric non-negative definite matrix. There exists some nxn matrix B such that

A

B’B=A

40
Q

Let A and B be nxn symmetrix matrices. If A is positive definite, then there exists a non-singular matrix Q, such that

A
  • Q’AQ=I and
  • Q’BQ=D

where D is a diagonal matrix whose diagonal elements are roots of det(B-lambda A) = 0

41
Q

If A and B are both non-negative dfinite, then there exists a matrix Q such that both Q’AQ and Q’BQ are both

A

diagonal

42
Q
A
43
Q

Consider a matrix of the attached form.

It can be shown that the inverse of this matrix is:

A
44
Q

Let A be an nxn matrix. A is idempotent if

A

AA=A

45
Q

If A is an nxn idempotent matrix with rank n, then

A

A=I

46
Q

If A is an nxn idempotent matrix of rank less than n, then A is

A

positive semidefinite

47
Q

If A is an nxn idempotent matrix with rank r, then it has ___ non-zero eigenvalues each equal to ___.

A

r non-zero eigenvalues

each equal to 1

48
Q

Let A ben an nxn (symmetric) idempotent matrix.

  • A’ is
  • Let P be an orthonormal matrix. P’AP is
  • Let P be an nxn non-singular matrrix. PAP-1 is
  • I - A is
A
  • A’ is (symetric) idempotent
  • Let P be an orthonormal matrix. P’AP is (symmetric) idempotent
  • Let P be an nxn non-singular matrrix. PAP-1 is idempotent
  • I - A is (symmetric) idempotent
49
Q

Trace A

A

Σ diagonal elements

50
Q

Let A be mxn and B be an nxm matrix.

Trace(AB) =

A

trace(BA)

51
Q

trace(ABC)

A

trace(CAB)

52
Q

Let A be an nxn matrix and P be a non-singlular nxn matrix

trace(A) =

A

trace(P-1AP)

53
Q

Let A be an nxn matrix and P be a orthonormal nxn matrix

trace(A) =

A

trace(A) = trace(P’AP)

54
Q

Let A be an nxn matrix with eigenvalues lambda 1, … n

trace(A) =

A

sum of eigenvalues

55
Q

If A is an idempotent matrix, then

trace(A) =

A

trace(A) = rank(A)

56
Q

Let A and B be nxn matrices, and let a and b be scalars.

trace(aA + bB) =

A

trace(aA + bB) = a*trace(A) + b*trace(B)

57
Q

Let A be an nxn matrix

trace(A) =

A

trace(A) = trace(A’)

Linear Models (2 decks)

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