Lecture Notes 1: Matrix Algebra Part C: Pivoting and Reduced
The Intermediate Matrices and Pivot Steps After k 1 pivoting operations have been completed, and column ‘ k 1 (with ‘ k 1 k 1) was the last to be used: 1 The rst or \top" k 1 rows of the m n matrix form a (k 1) n submatrix in row echelon form 2 The last or \bottom" m k + 1 rows of the m n matrix form an (m k + 1) n submatrix whose rst ‘
Appendix A - University of Texas at Austin
of each row is called a pivot, and the columns in which pivots appear are called pivot columns If two matrices in row-echelon form are row-equivalent, then their pivots are in exactly the same places When we speak of the pivot columns of a general matrix A, we mean the pivot columns of any matrix in row-echelon form that is row-equivalent to A
Theorem 6) The pivot columns of a matrix A form a basis for
Theorem (6) The pivot columns of a matrix A form a basis for the column space Col(A) Proof The proof has two parts: show the pivot columns are linearly independent and show the pivot columns span the column space We use the reduced echelon matrix of A in the proof We designate it by B If A and B have r pivot
Pivoting in Maple/Matlab/Mathematica
Here we have just called the pivot command, but did not save the output of the command into a variable If I check the value of the matrix A (by typing matrix(A) and pressing enter), I will see that its unchanged So let’s just recall the command again, this time storing the resulting matrix in a matrix B: B := pivot(A,1,1);
Lecture Notes 1: Matrix Algebra Part C: Pivoting and Matrix
Aunitriangularmatrix is a triangular matrix (upper or lower) for which all elements on the principal diagonal equal 1 Theorem The determinant of any unitriangular matrix is 1 Proof The determinant of any triangular matrix is the product of its diagonal elements, which must be 1 in the unitriangular case when every diagonal elements is 1
Excessive Pivot Ratios Cheatsheet - MSC Software
grid point id degree of freedom matrix/factor diagonal ratio matrix diagonal 11077 R2 1 84205E+12 2 06674E+03 1198 T3 -1 73615E+14 6 82135E+05
Solutions to Section 1 2 Homework Problems S F
3 x6 matrix and the last column of this augmented matrix cannot be a pivot column Since there are only three rows , there can be at most three pivot columns and we are told that these three pivot columns are among the first five columns 2 7 If the coefficient matrix of a linear system has a pivot position in every row , then
A quick example calculating the column space and the
Determine the column space of A = A basis for col A consists of the 3 pivot columns from the original matrix A Thus basis for col A = Note the basis for col A consists of exactly 3 vectors
[PDF] livre des merveilles du monde de marco polo fiche lecture
[PDF] le livre des merveilles marco polo texte intégral pdf
[PDF] la fameuse invasion de la sicile par les ours questionnaire de lecture
[PDF] la fameuse invasion de la sicile par les ours film
[PDF] mobilisation de connaissances ses exemple
[PDF] la fameuse invasion de la sicile par les ours résumé
[PDF] la fameuse invasion de la sicile par les ours fiche de lecture
[PDF] la fameuse invasion de la sicile par les ours analyse
[PDF] l autonomie en crèche
[PDF] exemple ec2
[PDF] le pianiste personnages principaux
[PDF] le pianiste résumé complet du film
[PDF] le pianiste personnages principaux livre
[PDF] methodologie ec1
Lecture Notes 1: Matrix Algebra
Part C: Pivoting and Matrix Decomposition
Peter J. Hammond
Autumn 2012, revised Autumn 2014
University of Warwick, EC9A0 Maths for Economists Peter J. Hammond 1 of 46Lecture Outline
More Special Matrices
Triangular Matrices
Unitriangular MatricesPivoting to Reach the Reduced Row Echelon FormExample
The Row Echelon Form
The Reduced Row Echelon Form
Determinants and Inverses
University of Warwick, EC9A0 Maths for Economists Peter J. Hammond 2 of 46Outline
More Special Matrices
Triangular Matrices
Unitriangular Matrices
Pivoting to Reach the Reduced Row Echelon Form
Example
The Row Echelon Form
The Reduced Row Echelon Form
Determinants and Inverses
University of Warwick, EC9A0 Maths for Economists Peter J. Hammond 3 of 46Triangular Matrices: Denition
Denition
A square matrix is
u pper (r esp. lo wer triangula r if all its non-zero o diagonal elements are above and to the right (resp. below and to the left) of the diagonal | i.e., in the upper (resp. lower) triangle bounded by the principal diagonal.IThe elements of an upper triangular matrixU
satisfy (U)ij= 0 wheneveri>j. IThe elements of a lower triangular matrixL
satisfy (L)ij= 0 wheneveriExercise
Prove that the transpose:
1.U>of any upper triangular matrixUis lower triangular;
2.L>of any lower triangular matrixLis upper triangular.Exercise
Consider the matrixEr+q
that represents the elementary row operation of adding a multiple oftimes row q to row r.Under what conditions isEr+q
(i) upper triangular? (ii) lower triangular?Hint:Apply the row operation to the identity matrixI.Answer:(i) iq
Products of Upper Triangular Matrices
Theorem
The productW=UVof any two upper triangular matricesU;V is upper triangular, with diagonal elements w ii=uiivii(i= 1;:::;n) equal to the product of the corresponding diagonal elements ofU;V.Proof.Given any two upper triangularnnmatricesUandV,
the elements (wij)nnof their productW=UVsatisfy w ij=( Pj k=iuikvkjifij0 ifi>j
becauseuikvkj= 0 unless bothikandkj.SoW=UVis upper triangular.
Finally, puttingj=iimplies thatwii=uiiviifori= 1;:::;n.University of Warwick, EC9A0 Maths for Economists Peter J. Hammond 6 of 46
Products of Lower Triangular Matrices
Theorem
The product of any two lower triangular matrices
is lower triangular.Proof.Given any two lower triangular matricesL;M,
taking transposes shows that (LM)>=M>L>=U, where the productUis upper triangular, as the product of upper triangular matrices.HenceLM=U>is lower triangular,
as the transpose of an upper triangular matrix. University of Warwick, EC9A0 Maths for Economists Peter J. Hammond 7 of 46Determinants of Triangular Matrices
Theorem
The determinant of any nn upper triangular matrixU equals the product of all the elements on its principal diagonal.Proof.Recall the expansion formulajUj=P
2sgn()Qn
i=1ui(i) where denotes the set of permutations onf1;2;:::;ng. BecauseUis upper triangular, one hasui(i)= 0 unlessi(i). SoQn i=1ui(i)= 0 unlessi(i) for alli= 1;2;:::;n.But the only permutation2 which satisesi(i)
for alli= 1;2;:::;nis the identity permutation. Because sgn() = 1, the expansion reduces to the single term jUj= sgn()Y n i=1ui(i)=Y n i=1uii which is the product of the diagonal elements, as claimed. University of Warwick, EC9A0 Maths for Economists Peter J. Hammond 8 of 46Inverting Triangular Matrices
SimilarlyjLj=Qn
i=1`iifor any lower triangular matrixL.Evidently:Corollary
A triangular matrix (upper or lower) is invertible if and only if no element on its principal diagonal is 0.In the next slide, we shall prove:Theorem
If the inverseU1of an upper triangular matrixUexists, then it is upper triangular.Taking transposes leads immediately to:Corollary
If the inverseL1of an lower triangular matrixLexists, then it is lower triangular. University of Warwick, EC9A0 Maths for Economists Peter J. Hammond 9 of 46Inverting Triangular Matrices: Proofs
Recall the (n1)(n1) cofactor matrixCrs
that results from omitting rowrand columnsofU= (uij). When it exists,U1= (1=jUj)adjU, so it is enough to prove that thennmatrix (jCrsj) of cofactor determinants, whose transpose (jCrsj)>is the adjugate, is lower triangular. In caserMore Special Matrices
Triangular Matrices
Unitriangular Matrices
Pivoting to Reach the Reduced Row Echelon Form
Example
The Row Echelon Form
The Reduced Row Echelon Form
Determinants and Inverses
University of Warwick, EC9A0 Maths for Economists Peter J. Hammond 11 of 46 Unitriangular Matrices: Denition and Two PropertiesDenition
A unitriangula r matrix is a tria ngularmatrix (upp ero rlo wer) for which all elements on the principal diagonal equal 1.Theorem The determinant of any unitriangular matrix is 1.Proof. The determinant of any triangular matrix is the product of its diagonal elements, which must be 1 in the unitriangular case when every diagonal elements is 1. University of Warwick, EC9A0 Maths for Economists Peter J. Hammond 12 of 46 Converting a Diagonal Matrix to Unitriangular FormTheorem
SupposeUis any upper triangular matrix
with the property that all its diagonal elements u ii6= 0.Then there exists a diagonal matrixD
such that bothDUandUDare upper unitriangular.Similarly for any lower triangular matrixL
with the property that all its diagonal elements`ii6= 0.University of Warwick, EC9A0 Maths for Economists Peter J. Hammond 13 of 46
Converting a Diagonal Matrix: Proof
DeneDas the diagonal matrixdiag((1=uii)ni=1)
whose diagonal elementsdiiare the reciprocals 1=uii of the corresponding elementsuiiof the upper triangular matrixU, all of which are assumed to be non-zero. ThenDUis upper unitriangular because (DU)ik=diiikand so (DU)ij=X n k=1diiikukj=diiuij=(1 wheni=j;
0 wheni>j:
The same holds forUDwhose elements (UD)ij=uijdjj
are also 1 wheni=jand 0 wheni>j.University of Warwick, EC9A0 Maths for Economists Peter J. Hammond 14 of 46
The Product of Unitriangular Matrices Is UnitriangularTheorem
The productW=UVof any two upper unitriangular nn
matricesUandVis also upper unitriangular.Proof.Because bothUandVare upper triangular, so isW=UV.
Also, eachielement of the principal diagonal ofWiswii=uiivii, which is 1 because unitriangularity implies thatuii=vii= 1. It follows thatWis upper unitriangular.The same argument can be used to show that the product of any two lower unitriangularnnmatrices is also lower unitriangular. University of Warwick, EC9A0 Maths for Economists Peter J. Hammond 15 of 46 The Inverse of a Unitriangular Matrix Is UnitriangularTheorem
Any upper unitriangular nn matrixUis invertible,
with an upper unitriangular inverseU1.Proof. BecauseUis unitriangular, its determinant is 1, soV=U1exists.BecauseUis upper triangular, so isU1.
Alsouiivii=ii= 1 for alli= 1;2;:::;n,
implying thatvii= 1=uii= 1. ThereforeU1is indeed upper unitriangular.The same argument can be used to show that the inverse of any lower unitriangularnnmatrix is also lower unitriangular. University of Warwick, EC9A0 Maths for Economists Peter J. Hammond 16 of 46Outline
More Special Matrices
Triangular Matrices
Unitriangular Matrices
Pivoting to Reach the Reduced Row Echelon Form
Example
The Row Echelon Form
The Reduced Row Echelon Form
Determinants and Inverses
University of Warwick, EC9A0 Maths for Economists Peter J. Hammond 17 of 46Three Simultaneous Equations
Consider the system
of three simultaneous equations in three unknowns, which depends upon two \exogenous" constantsaandb: x+yz= 1 xy+ 2z= 2 x+ 2y+az=b It can be expressed as using an augmented 34 matrix: 1 111 11 22 1 2ab or, perhaps more usefully, a doubly augmented 37 matrix:1 1111 0 0
11 220 1 0
1 2ab0 0 1
whose last 3 columns are those of the 33 identity matrixI3. University of Warwick, EC9A0 Maths for Economists Peter J. Hammond 18 of 46The First Pivot Step
Start with the doubly augmented 37 matrix:
1 1111 0 0
11 220 1 0
1 2ab0 0 1
First, we
pivot ab outthe element in ro w1 and column 1 to zeroize the other elements of column 1. This elementa ryro wop eration requ iresus to subtr actro w1 from both rows 2 and 3. It is equivalent to multiplying by the lo wertriangula r matrix E1=0 @1 0 0 1 1 01 0 11
A Note that is the result of applying the same row operation toI.The result is:
1 1111 0 0
02 321 1 0
0 1a+ 1b11 0 1
University of Warwick, EC9A0 Maths for Economists Peter J. Hammond 19 of 46The Second Pivot Step
After augmenting again by the identity matrix, we have:1 1111 0 01 0 0
02 321 1 00 1 0
0 1a+ 1b11 0 10 0 1
Next, we pivot about the element in row 2 and column 2.Specically, multiply the second row by12
then subtract the new second row from the third to obtain:1 1111 0 01 0 0
0 132 1212 12 0012 0
0 0a+52b12
3212 10 12 1quotesdbs_dbs19.pdfusesText_25