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MA 405 > test 1 vocab > Flashcards

test 1 vocab Flashcards

1
Q

equivalent

A

when two systems possess equal solutions

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2
Q

unique solution

A

there is one and only one set of values for the xi’s that satisfies all equations simultaneously

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3
Q

no solution

A

there is not set of values for the xi’s that satisfies all equations simultaneously- the solution set is empty

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4
Q

infinitely many solutions

A

there are infinitely many different sets of values for the xi’s that satisfy all equations simultaneously. It is not difficult to prove that if a system has more than one solution, then it has infinitely many solutions.

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5
Q

elementary operations

A
  1. ) interchanging ith and nth equations
  2. )replace ith equation by nonzero multiple of itself
  3. ) replace jth equation by a combination of itself plus a multiple of the with equation
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6
Q

square system

A

n equations and n unknowns

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7
Q

Gaussian eliminations

A

1.) eliminate all terms below the first pivot
2.) select a new pivot
3.) eliminate all terms below second pivot
and continue

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8
Q

triangularized

A

all pivots are 1

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9
Q

back substitution

A

the last equation is solves for the value of the last unknown and then substituted back into the penultimate equation and so on

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10
Q

scalar

A

a real or complex number

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11
Q

row

A

horizontal line

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12
Q

column

A

vertical line

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13
Q

submatrix

A

of A is an array obtained by deleting any combination of rows and columns from A

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14
Q

shape or size

A

m(rows)xn(columns)

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15
Q

Gauss-Jordan Method

A
  1. ) at each step, the pivot element is forced to be 1

2. ) all terms above and below the pivot are eliminated

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16
Q

tridiagonal

A

the nonzero elements occur only on the sub diagonal, main diagonal, and super diagonal

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17
Q

partial pivoting

A

at each step, search the positions on and below the pivot position for the coefficient with the maximum magnitude. If necessary interchange the rows to bring the larger number into the pivot position

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18
Q

row scaling

A

multiplying selected rows by non zero multipliers

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19
Q

column scaling

A

multiplying selected columns by nonzero multipliers

-alters exact solution

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20
Q

complete pivoting

A

search the pivot position and every position below or to the right for the maximum magnitude, if necessary perfume row and column interchange to bring largest number to pivot position

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21
Q

ill conditioned

A

some of the perturbation in the system can produce relatively large changes in the exact solution

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22
Q

well conditioned

A

if not ill conditioned

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23
Q

rectangular

A

if m and n are no the same

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24
Q

main diagonal

A

where the pivot positions are located, the diagonal line from the upper left hand to the lower right hand corner

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25
Q

row echelon form

A
  1. )if Ei, consists entirely of zeros, then all rows below Ei are also entirely zero, i.e. all zero rows are at the bottom
  2. ) if the first nonzero entry in Ei* lies the nth position then all entries below the ith position in columns E1, E2…E*j are zero
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26
Q

rank

A

in echelon form, the number of pivots, number of nonzero rows in E, number of basic columns in A

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27
Q

basic columns

A

those columns in A that contain the pivotal positions

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28
Q

reduced row echelon form(EA)

A
  1. ) E is in row echelon form
  2. ) the first nonzero entry in each row(each pivot) is 1
  3. )all entries above each pivot are 0
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29
Q

consistent

A

a system of m linear equations in n unknowns that posses at least one solution

  • rank[A|b]=rank(A)
  • b is a nonbasic column in [A|b] or is a combination of the basic columns in A
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30
Q

inconsistent

A

a system of m linear equations in n unknowns that has no solutions, when a row of all zeros produces a nonzero solution

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31
Q

homogeneous system

A

the right hand side consists entirely of 0’s

-consistency is never an issue because the zero solution is always a solution

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32
Q

nonhomogeneous system

A

there is at least one nonzero number on the right hand side

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33
Q

trivial solution

A

the solution consisting of all zeros

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34
Q

basic variables

A

when there are more unknowns then equations, we have to pick “basic” unknowns and solve for these in terms of the other unknowns
- there are r basic variables

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35
Q

free variables

A

whose values must remain arbitrary or free

- there are n-r free variables

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36
Q

general solution

A

use Gaussian elimination to reduce to row echelon form. identify basic and free variables. apply back substitution and solve for the basic variables in terms of the free variables
x=P(particular solution)+xf1h1 +xf2h2+…

37
Q

equal matrices

A

when A and B are the same size and corresponding entries are equal

38
Q

column vector

A

an array consisting of a single column

39
Q

row vector

A

an array consisting of a single row

40
Q

addition of matrices

A

if A and B are mxn the sum is the mxn matrix A+B, by adding corresponding entries

41
Q

additive inverse (-A)

A

the matrix obtained by negating each of the entries

42
Q

transpose (A^T)

A 
of a mxn matrix, is the nxm matrix A^T obtained by interchanging rows and columns.
A^T= aji
properties:
(A+B)^T= A^T+B^T
(sA)^T=sA^T
43
Q

conjugate matrix (A^-)

A

a^-ij,

44
Q

conjugate transpose (A^-T)(A*)

A

a^-ji
properties:
(A+B)* =A+B
(sA)= s^-A

45
Q

diagonal matrix

A

entries are symmetrically located about the main diagonal

46
Q

symmetric matrix

A

A =A^T, when aij=aji

47
Q

skew-symmetric matrix

A

A=-A^T, when aij= -aji

48
Q

hermitian matrix

A

A=A*, when aij=a^-ji. the complex analog of symmetry

49
Q

skew-hermitian matrix

A

A=-A*, when aij= -a^-ji. the complex analog of skew symmetry

50
Q

linear function

A
  1. ) f(x+y)= f(x)+f(y)

2. ) f(sx)=sf(x)

51
Q

conformable

A

in AB when A has exactly as many columns as B has rows

52
Q

matrix product

A

for comfortable matrices Amxp= aij and Bpxn=bij, AB is the mxn matrix whose i,j entry is the inner product of the ith row of A with the jth column in B
-matrix multiplication is NOT commutative

53
Q

cancellation law

A

when sB=sY and s=/0 implies B=Y

54
Q

linear system

A

Ax=b

55
Q

distributive and associative laws

A

for comfortanble matrices:
A(B+C)=AB+BC
(D+E)F=DF+EF
A(BC)=AB(C)

56
Q

identity matrix (I)

A

nxn matrix with 1’s on the main diagonal and 0’s everywhere else
AIj=Aj

57
Q

reverse order law for transposition

A

for comfortable A and B
(AB)^T=B^TA^T
(AB)=BA*

58
Q

Trace

A

for a square matrix, is the sum of its main diagonal entires
trace(AB)=trace(BA)

59
Q

block matrix multiplication

A

A and B are partitioned into sub matrices, referred to as blocks. if the pairs (Aik, Bkj) are comfortable then A and B are comfortably partitioned

60
Q

reducible systems

A

block-triangular systems

61
Q

inverse of A

A

given, A and B are square, AB=I and BA=I

the inverse of A is, B=A^-1

62
Q

nonsingular matrix

A

an invertible square matrix

63
Q

singular matrix

A

a square matrix with no inverse

64
Q

matrix equations

A

if A is nonsingular then there is a unique solution for X, (Anxn)(Xnxp)=Bnxp and the solution is:
X=A^-1(B)
for system of n linear equations and n unknowns:
(Anxn)(Xnx1)=bnx1
X=A^-1(b)

65
Q

existence of an inverse

A

for nun the following are equivalent:

  • A^-1 exists (nonsingular)
  • rank(A)=n
  • A——>(Guass Jordan)—>I
  • Ax=0 (implies x=0)
66
Q

computing the inverse

A

[A|I}—->GJ—>[I|A^-1]

67
Q

properties of matrix inversion

A 
For nonsingular A and B:
(A^-1)^-1=A
AB is nonsingular
(AB)^-1=B^-1A^-1
(A^-1)^T= (A^T)^-1
(A-1)*=(A*)^-1
68
Q

Sherman-Morrison Formula

A

if Anxn is nonsingular and c and d are nx1 columns such that 1+d^TA^-1c is not 0, then the sum of A+cd^T;
(A+cg^T)^-1= A^-1 - (A^-1cd^TA^-1)/(1+d^TA^-1c)

69
Q

sherman morrison woodbury formula

A

if C and D are nxk such that (I+D^TA^-1) exists then:

(A+CD^T)^-1 = A^-1C(I+D^TA^-1C)^-1D^TA^-1

70
Q

Neumann Series

A

if lim n—>infinity, then I-A is nonsingular and
(I-A)^-1=I+A+A^2… sum of A^K.
provides approximation of (I-A)^-1 when A has entires of small magnitude.

71
Q

ill conditioned

A

if a small relative change in A can cause a large relative change in A^-1.

72
Q

condition number

A

how the degree of ill conditioning is gauged.

k=||A|| ||A^-1|| where ||*|| is a matrix norm.

73
Q

sensitivity

A

of the solution of Ax=b to perturbations(or errors) in A is measured by the extent to which A is an ill conditioned matrix.

74
Q

elementary matrices

A

matrices in the form I-uv^T, where u and v are nx1 columns such that v^tu=/1
they are nonsingular and
(I-uv^T)^-1 = I- (uv^T)/(v^Tu-1)

75
Q

type 1

A

interchanging rows

76
Q

type 2

A

multiplying rows(columns) by a scalar

77
Q

type 3

A

adding a multiple of a row(column) i to a row(column)j

78
Q

products of elementary matrices

A

A is a nonsingular matrix if and only if A is the product of elementary matrices of type 1,2, and 3

79
Q

equivalent matrices(~)

A

when B can be derived from A by a combination of elementary row and column operations A~B

80
Q

row equivalent (~row)

A

when B can be obtained from A by preforming a sequence of elementary row operations only A~rowB

81
Q

column equivalent (~col)

A

when B can be obtained from A by preforming a sequence of elementary column operations only A~colB

82
Q

transitive

A

A~B and B~C——> A~C

83
Q

Rank normal form (Nr)

A

A~Nr= (Ir 0)

(00)

84
Q

LU factorization of A

A

A=LU, product of lower triangle matrix L and an upper triangle matrix U.
the decomposition of A into A=LU
- matrices L and U are called LU facts of A

85
Q

elementary lower triangular matrix

A

Tk = I-cke^Tk, where ck is a column with zeros in the first k positions

86
Q

leading principle sub matrices

A

the sub matrices taken from the upper left hand corner

87
Q

positive definite

A

A symmetric matrix A possessing an LU factorization in which each pivot is positive

88
Q

band matrix

A

aij=0

89
Q

bandwidth

A

when |i-j|>w for some positive integer w

MA 405 (3 decks)

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