matrix and determinant
TRANSCRIPT
Halil Aydemir - 19.06.2011
Matrix and Determinant
matrix and matrices
rows and columns
square matrix
entries
main diagonal
diagonal matrix
scalar matrix
zero matrix
identfy matrix
triangular matrices
matrix addition
matrix multiplication
transpose
symetric matrix
skew-symetric matrix
reduced row echelon form
linear systems
gouss-jordon reduction
inverse of matrix
trace of matrix
determinant
cramer's rule
adjoint
rank
subspace
linear combination
spans
linearly independent/dependent
basis
dimension
eigenvalues , eigenvectors and diagonalization
characteristic polynomail
coyley–hamilton theorem
Halil Aydemir - 19.06.2011
Matrix and Matrices
Definition: An mxn matrix A is a rectangular array of real number arranged in “m” ,
horizontally rows and “n” vertical columns.
A[ ]
- : the (i,j) so this mean entry of A.
-“m” and “n” are the integers.
-m: the number of rows of A
-n: the number of columns of A
*if m=n that A is called a square matrix.
Example:
A is a 3x3 matrix with entries; =1 , =0 ...
B is a 2x2 matrix with entries; =-1 , =0 ...
C is a 1x1 matrix with entries; =7
D is a 3x1 matrix with entries; =2 , =0 ...
Definition: Let A be an nxn matrix;
........... The entries , ..... are called
A= .......... the main diagonal entries.
. . .
. . .
.......... the main diagonal
Halil Aydemir - 19.06.2011
Definition: Let A=[ ] be a square matrix if =0 for i j then A is called a diagonal
matrix.Its shape of a diagonal matrix is of the form;
...........
A= ..........
. . .
. . .
.......... the main diagonal
but
is square but not diagonal because of entries.
also
diag(0,0,0)=
or diag(2,-1,3)=
Definition: A diagonal marix A[ ] is called a scalar matrix if = ...= =c
(c is constant)
Example:
Definition: Let A=[ ] be an mxn matrix if all entries of A are zero than A is called zero
matrix.Zero matrix is donated by 0.
Example:
Definition: If all entries on the main diagonal are “1” than a diagonal matrix is an identfy
matrix.This matrix is donated by .
Example:
=
=
Halil Aydemir - 19.06.2011
Definition: A square matrix A[ ] is said to be upper triangular if =0 for all i>j.İts shape
of an upper triangular matrix is of the form.
...........
A= ..........
0 . ..............
0 0 ...........
0 0..........
Definition: A square matrix A[ ] is said to be lower triangular if =0 for all i<j.İts shape
of an lower triangular matrix is of the form.
0...............0
A= .............0
. . ..............0
. . ............0
..........
*If the matrix is lower or upper matrix,it must be diagonal and square matrix.
Definition: Let A[ ] and B[ ] be two mxn matrices A and B are equal , if = for all
i,j.
Example: The matrices A=
and B=
so x=0,y=5 and z=1.
Matrix Addition
The sum of mxn matrices A=[ ] and B[ ] is define as the mxn matrix C[ ] with entries
= + for all i,j.The sum of two matrices A and B is defined only when A and B are the
same size.
Halil Aydemir - 19.06.2011
Properties of Matrix Additions
Let A,B,C are mxn matrices;
1)A+B=B+A
2)A+(B+C)=(A+B)+C
3)For each mxn matrix A,there is a unique matrix B such that A+B=B+A=0 for mxn matrix.B
is donated by A and called the negative of A.
Multiplication by Scalars
Let A=[ ] be an mxn matrix.The matrix rA is defined as the mxn matrix B=[ ] with
entries.
Properties of Multiplication of Matrix
A,B matrices and r,s is real number;
1)r(A+B)=rA+rB
2)(r+s)=(rA).B=A(rB)
4)r(sA)=(rs)A=s(rA)
Definition: If , .... are mxn matrices and , ... are real numbers than
+ ....+ is called linear combination of , .... and , ... are called
coefficients.
Example:
and
then find the linear combination –A+3/2B.
-A+3/2B =
Matrix Multiplication
Definition: Let A be an mxp matrix and B be a pxn matrix.The product AB= the mxn
matrix given by = .
Note: If two matrices product,their size must be balance.
A=mxp
B=nxp so AB must be mxn size.
Halil Aydemir - 19.06.2011
Example:
,
find AB and BA.
A . B B . A
2x3 = 3x3 A.B=
3x3 2x3 so B.A is undefined.
2x3
Example:
Let
and
so compute A.B
A.B
3x2 2x2
3x2 A.B=
Example:
Let
and
if A.B=
find “x” and “y”.
2x+2y+18=12 , x=1
4-y+6=6 , y=4
Example:
Let and
Find AB and BA.
and
Halil Aydemir - 19.06.2011
Example:
Let
,
compute A.B
Example:
Let A =
, B=
and C=
compute A.B and A.C
A.B=
, A.C=
so A.B=A.C
Note: a,b,c R , a.b=a.c for a,b,c R and a 0 that b=c , but it’s not true fro matrices
A.B=A.C and A 0 whereas B C.
Properties of Matrix Multiplication:
If A,B,C are of the appropriate sizes then;
1) A(BC)=(AB)C
2) A(B+C)=AB+AC and (B+C)A=BA+CA
Remark: If A is an mxn matrix and then like;
A. =
Definition: If A=[ ] is an mxn matrix and the transpose of A is the nxm matrix
given for all i,j.
Example:
A=
, B= , C
= Compute .
,
,
Halil Aydemir - 19.06.2011
Properties of Traspose:
1)
2)
3) r
4) !!!
If P and A is square matrix , we can define the powers of A as follows;
=A.A.A......A (p times)
Definition: If =A then A is symetric matrix.
If =-A then A is skew-symetric matrix.
1) A, are of the same size.
2) A is symetric if an only if for all i,j.
3) A is skew-symetric if an only if for all i,j
Definition: If an mxn matrix A satisfied the following properties then it is said to be in
reduced row echelon form.
a)All zero rows , if there are any appear at bottom of the matrix A.(bottom of the matrix must
be zero)
b)Each leading entry is one. (All rows must be start with 1)
c)For each nonzero row,the leading one appear to the right and below my leading one’s in
preceding.(other ones must be any right side of the first row one.)
d)Each leading one is the unique nonzero entry of its own column.(All columns , which has
leading , must be one or zero.)
Note: If the matrix provide first three part , it will called row echelon matrix.İf the matrix
provide all of the parts , it will called reduce row echelon matrix.
Example:
,
,
Halil Aydemir - 19.06.2011
,
,
,
,
A is not row echelon form and also not in reduce row echelon form.
B is row echelon form and also reduce row echelon form .
C is row echelon form and also reduce row echelon form .
D is row echelon form and also reduce row echelon form .
E is not row echelon form and also not in reduce row echelon form.
F is not row echelon form and also not in reduce row echelon form.
G is row echelon form but not in reduce row echelon form.
H is not row echelon form and also not in reduce row echelon form.
T is row echelon form and also reduce row echelon form .
Definition: Any one of the following operations is called a elementary row operation on an
mxn matrix of A.
1) Interchange row î and row j.
2) Multiply row î by a nonzero scalar k.
3) Add k times row î to row j.(î j)
Definition: Let A and B be mxn matrices.A is row equivolen to B if B can be obtained by
applying a finite squence of elementary row operations to A.
Example:
and
conver A to B.
A~B
-2
2 -1
2
Halil Aydemir - 19.06.2011
Theorem:
i)Every matrix is row equivalent to itself.(A~A)
ii)If A is row equivalent to B then B is row equivalent to A.(A~B so B~A)
iii)If A is row equivalent to B and B is row equivalent to C then A is now equivalent to
C.(A~B,B~C so A~C)
Note: Every mxn matrix is row equivalent to a unique matrix in reduced row echelon
form.This matrix is called the reduced row echelon form of the matrix.
Example:
Let
find the row echelon form of A.
~
~
row echelon and reduce row echelon
Linear System
A linear system of m equations in m unknowns
. . . .
. . . .
. . . .
, : constants
1
2
-2
1
2
-1
2
Halil Aydemir - 19.06.2011
is called the coefficient matrix of the linear system.
,
this system can eb written in matrix form as follow;
AX=B
Gouss-Jordon Reduction
StepI : Form the augmented matrix [A|B].
Step II : Obtain the reduced row echelon form [C|D] of the augmented matrix
[A|B].
Step III : For each nonzero row,solve the corresponding equation for the
unknown associated with leading one in that row.
Definition: The augmented matrix is the mx(n+1) matrix it can be obtained by adjoining the
column B to A is donated by [A|B].
. . . .
. . . .
. . . .
[A|B]=
Example:
Solve the linear system x+2y+3z=9 by Gouss-Jordon reduction.
2x – y +z=8
3x -z =3
Halil Aydemir - 19.06.2011
The augmented matrix is;
Step I :
Step II :
~
~
~
~
~
~
Step III : 1.x+0.y+0.z=2 so x=2
0.x+1.y+0.z=-1 so y=-1 and z=3
Example:
Solve the system x + y + 2z - 5w = 3 by Gouss-Jordon reduction.
2x + 5y - z - 9w= -3
2x + y - z + 3w = -11
x - 3y + 2z + 7w = -5
~
~
-2
-1/5
6
-1/4
-3
-1
-2
-1 1/4
-2
-7
-2
Halil Aydemir - 19.06.2011
~
~
~
x + 2w = -5 so x = -5 – 2w
y – 3w = 2 so y = 2 +3w
z – 2w = 3 so z= 3+2w
Note: If the last column has a leading entry than the system has no solution.This is called
inconsisted.Otherwise it is called consisted.If system is constant and it has no free variable
then it has a unique solution.
Theorem: Let AX=B and CX=D so if [A|B] is row equivalent to [C|D] then AX=B and
CX=D have the same solution.Corollary , if A~C then AX=C and CX=0 have the same
solution.
Homogenius System
A linear system fo the form;
. . . .
. . . .
is called a homogenious system.A homogenius
system can be written in matrix form as AX=0 ,
-1/5
-2
-1
-2
4
-2
-2
-2
-1
-2
Halil Aydemir - 19.06.2011
is a solution of the homogenius system.This solution is called
the trivial solution of the homogenius system.Other solutions of the given system are called
nontrivial solutions.
Example: Consider the homogenious system ;
~
~
~
~
so x=0 , y=0 , z=0 ; x=y=z is a unique solution of theorem.The given system has only trivial
solution.
Example: Solve the homogenious system;
~
~
~
~
x+w=0 , y-w=0 , z+w=0 so x=-w , y=w , z=-w , w =free so this mean so many solutions.
w=r in any real number.Thus the solution is x=-r , y=r , z=-r , w=r we can be assigned any
real number than the given system has infinitely many solutions and
general solution is
=
and solution set is
=
-2
1
-2
-2
-2
1/5 3
-1
-1
-1
-2 -1
-1
-1
-1
-1
-1
-1
Halil Aydemir - 19.06.2011
Example: Find all solutions to the given linear system.
~
~
~
~
is in reduced row echelon form and are basic but is free variable;this system
has infinity many solution , , , , so;
the general solution is
and solution set is
Example: Find all solutions to the given linear system.
~
~
-3
-2
-1/2
-2
-1/2
-2
-2
-2 -2
-2
-1
-2
-2
-2
-3
-2
6
-2
+1
-2
Halil Aydemir - 19.06.2011
are basic but is free so so
so
the general set is
and solution set is
Example: Solve the given system;
~
~
~
. so this system has no solution.
Example: Find conditions that the b’s satisfy for the system consistant;
~
the given system is consistant if and only if we have and
x2
-2 -3
-2
1/6
-2
-3
-2
8
-2
2
-2
-4
-2
3
-2
-1
-2
1
-2
-1
-2
Halil Aydemir - 19.06.2011
Example: Find all values of a for which the resulting system has ;
a)no solution b)a unique solution c)infinitely many soltions
~
~
~
=0 then are critical point
if a=3 then
is free so th system has infinity many solution for a=3.
if a=-3 then
-6 so it has no solution for a=-3.
if then
~
so
given system has unique solution for
Example: Find all values of a for which the resulting system has ;
a)no solution b)a unique solution c)infinitely many soltions
-1
-2
-2 -1
-2
-1
-2
-1/2
-2
Halil Aydemir - 19.06.2011
~
~
(a+2)(a-1)=0 so a=-2 and a=1 are critical point
if a=-2 then
0 3 it has no solution
if a=1 then
is basic but are free so the given system has infinity
many solutions.
if then
~
this system has unique solution.
Example: Let A and B symetric matrices;
a)show that A+B is symetric
b)show that A.B is symetric if and only if A.B=B.A
proof: , since A and B are symetric and if A+B symetric , A and B must be
equal to transpoze.
check that a)
A+B is symetric matrix.
b)
so
so A.B is symetric matrix
-1
-2
-2
-(1+a)
-2
-2
-a
-2
-2
-1
-2
-2
-2
-2
-2
-2
Halil Aydemir - 19.06.2011
The Inverse of A Matrix
Let A be an mxn matrix and A is called invertible (non-singular) if there exist a matrix B
such that A.B = B.A = I. The matrix B is called the inverse of A .
Theorem: The inverse of a matrix if there exist is unique . Proof;
Support that B and C are inverses of A ( B=?C)
A.B =B.A =I and A.C=C.A=I since B and C are inverses of A so B=B.I=B.(A.C)=(B.A).C
=I.C=C then B=C.
Note: The inverse of A is donated by like A. = .A=I
Example: Let
find the .
A is invertible.
Note: For invertible matrix must be definitelly square matrix.
Remark: Not every matrix has an inverser.
Example:
find
Properties of the Inverse
a) If A is invertible then is invertible and
b) If A and B are invertible matrices then A.B is invertible and
c) If A is invertible then is invertible and .
Proof of a: since A is invertible , write A. = .A=I . We have to find a matrix B such
that .B=B. =I so B=A since the inverse is unique.Therefore,
Halil Aydemir - 19.06.2011
Proof of b: It must be equal to right and left multiply. and such that
A. = .A=I and B. = .B=I .
From right multiply ; (A.B)( . )=A( .B) =A.I. =A. =I
From left multiply ; ( . )(A.B)= .A).B= .I.B=B. =I so they are equal.
Proof of c: From right ;
From left ;
the inverse of is so .
How to Find :
#Step1 : write [A|I]
#Step2 : obtain the reduced row echelon form [C|D] of [A|I].
#Step3 : if we obtain a zero row in the first part of matrix [C|D] then does not exist.If we
obtain an identify matrix I in the first part of matrix [C|D] then A is invertible and
=D.
Example:
[A|I]=
Note: [A|I] ~.....~.....~.....[C|D]
Example: [A|I]=
=?
does not exist invert , C is not invertible.
Definition: The trace of an nxn matrix A is desined by
Show that ;
a) where C is a scalar.
b)
c)
-1
-1
1/2
-3/2
-2
Halil Aydemir - 19.06.2011
Proof of a : Let then so
Proof of b : Let then so then
Proof of c : so then so
so
Note: Tr(A.B)=Tr(B.A)
Example: Find the inverse of the given matrices of a,b,c but if they are possible.
a)
~
~
so
b)
~
non invertible because of zeros.
Determinant
Let be an nxn matrix. is the (n-1)x(n-1) submatrix obtained by deleting the
row and column of A. is called the minor of .
is called the cofactor of
expansion of |A| a long the row and same for
Example: If
-1
-1 -1
1/2
-1
3
-2
Halil Aydemir - 19.06.2011
Example:
Example:
Example:
by using the first column expansion.
Example: If
fiind the determinant of te following matrices
Note: Change the place in determinant is having (-) .If multiply with some number of row
echolon , you must multiply with this number of determinant.If mutliply some row and add
other row , this calculate is not affect the determinant result.
Halil Aydemir - 19.06.2011
Example: Compute the determinant
.
Example: Compute the determinant
.
Example: Compute the determinant
.
Example: Compute the determinant
.
Example: Let
,
then
Example:
=
Example: Without expanding , show that
-4/5
1 1
Halil Aydemir - 19.06.2011
Example: Without expanding , show that
Properties of Determinants
1) .
2) If A has a zero row or column then .
3) If B is obtained from A by multiplying a row or column of A by k then .
4) If two rows or column of A are equal then .
5) If B is obtained from A by interchancing two rows or columns of A then .
6) If B is obtained from A by adding to each elementary of the row or column of A a
constant k times corresponding element the row or column of A then .
7) If a matrix is upper or lower triangular then .
Corollary: The determinant of diagonal matrix is the product of entries on the main diagonal.
8) .
Corollary: If A is nonsingular(A is invertible) then and
.
Cramer’s Rule: Let A be an nxn matrix , Cramer’s rule for solving the linear system AX=B
is as follows :
#Step 1: Compute |A|. If |A|=0 then Cramer’s rule is not appliciable.Use Gauss-Jordan
reduction.
#Step 2: If |A| 0 then for each i ,
. is the matrix obtained from A by replacing the
column of A by B.
Example: Consider the following linear system ;
- 1 - 1
Halil Aydemir - 19.06.2011
Example: |A| = 4 find | | and | |.
Example: If A & B are nxn matrices with |A| = 2 and |B| = -3 calculate | |.
Example:
without directly evulating.
Example: Prove that identify without evulating the determinants.
Note: Don’t do anything for calculating column or row.
+1
-1
+1
-1
Halil Aydemir - 19.06.2011
Definition: Let A=[ ] be an Axn matrix then the matrix
is
callded the adjoint of A.
Example: Let
compute the adjoint of A.
Theorem: If A=[ ] is an nxn matrx then A(Adj(A))=(Adj(A)).A=|A|.
Corollary: If A is an nxn matrix and |A| 0 then
.
Proof: Adj(A) = (Adj(A))=|A|. & |A| 0.
since A has a unique solution.
Theorem: A matrix is nonsingular iff |A| 0.
Corollary: For an nxn matrix A , AX=0 has a nontrival solution iff |A|=0.
Example: If possible solve the following linear system by cramer’s rule.
Halil Aydemir - 19.06.2011
Example: Use Cramer’s rule to find all values of a for which the linear system.
has the solution in which y=1.
Example: Prove the identify without evulating the determinants.
Example: Prove the identify without evulating the determinants.
Example: Without directly evulating , show that;
a
b c
Halil Aydemir - 19.06.2011
Example: Find the determinant of
Example: The inverse of certain matrix A is given
use this
information to find |A| and Adj(A).
Example: Let
a) By using sutable elementary row and column operations as well as row and column
expansions , show that |A|=6.
b)Find (2,2) and (3,1) entries of .
-2
-2
1
1
1
Halil Aydemir - 19.06.2011
a)
b)Transpose of (2,2) is (2,2) so
Transpose of (3,1) is (1,3) so
Note: When take determinant in adjoint , do not write entry!
Example: Find component “t” of the solution vector for following linear system.
-2
-1 -1
-1
-1
-2
Halil Aydemir - 19.06.2011
Example: Without expanding , show that ;
Example: What must be “k” if the matrix
is not invertible.
If it is not invertible so it must be determinant=0
.
Example: Solve the system by using the inverse of the coefficient matrix
-1
-1
-1
-1
-2
Halil Aydemir - 19.06.2011
Definition: A submatrix of A is any matrix obtained from A by deleting some rows , columns
of A , A is also considered to be a submatrix of A.
Definition: A nonzero matrix A is said to have rank r if atleast one of r-square submatrix is
nonsingular while every (r+1)-square submatrix of A is singular. A zero matrix is said to have
rank 0.
Example:
compute the rank of A.
How to Compute the Rank of Any Matrix
#Step1: Obtain the row echelon form B of A.
#Step2: The rank of A is the number of nonzero rows of B.
Some Knowledge about the Rank of a Matrix
Let A be an nxn matrix
1) A is nonsingular if and only if rank A=n.
2) Rank A=n if and only if
3) AX=B has a unique solution for every nx1 matrix B if and only if rank A=n.
4) AX=0 has a nontrivial solution if and only if rank A<n.
5) Elementery row operations don’t alter the rank of a matrix.
6) Equivalent matrices have the same rank.
-2
-3 -1
Halil Aydemir - 19.06.2011
List of Nonsingular Equivalent
TFAE for an nxn matrix A
1) A is nonsingular
2)
3) Rank A = n
5) AX=0 has only trivial solution
6) AX=B has a unique solution.
7) A is row equivalent to I
Example:
compute the rank of A.
Rank of A is 3 and
Example:
compute the rank.
Rank is 3 and
Example:
compute the rank.
Rank is 2 and
-3
-2 -1/3
-1/4
-1
-1
-3
-2
-1
-2
1 -1
Halil Aydemir - 19.06.2011
Subspace
Definition: An n-vector (or n-type) is an nx1 matrix.
row vector.
whose entries are real numbers called the components of x
: the set of all n-vectors.
: is called the n-spaces.
Definition: A no empty subset V of is a subspace of the following properties is satisfied.
1) If then .
2) If , then .
Note: 1)
2)
3)
Note: * , {0} are subspace of
* {0} is called the zero subspace of
* If V is a subspace then
Example: Consider the subset V of consisting of all vector of the form
, show
that V is a subspace of .
1)
2) Let
3) Let
1,2,3 is satisfied so subspace of .
Halil Aydemir - 19.06.2011
Example: Consider the subset V of consisting of all vector of the form
.Is V a
subspace of ?
so V is
not subspace.
Note: If V is a subspace then 0 V.
Example: Let A be an mxn matrix and consider the linear system The set V of all
solutions to this system is a subset of .Then V is a subspace of
All s is true so this system is subspace of .
Definition: A vector V is is said to be linear combination of vector if it can
be written as where are constants (real numbers).
Example: Let
The vector V is a linear combination of
and .We must find constants.
,
1
-1
1/6
-2 -2
Halil Aydemir - 19.06.2011
so V is a linear combination of , and V=2 .
Definition: Let S={ , } be set of vectors in a subspace V of . The set S spans
V or V is spanned by S if every vector in V is a linear combination of vector in S.
Example: Let S={ , } where
Determine whether S
spans .
X= ,
Example: Let S={ } where
Determine whether S spans
If then this system has no solution since a solution to this system can be
obtained for any choice of . Therefore , S does not span .
Definition: Set S={ } be a set of vectors in a subspace V of . The set S is
said to be linearly dependent , if we can find constants not a ll zero such that
0= . Otherwise S is linearly independent that is , S is linearly
independent if equation ca be satisfied only with
Example: Consider the vectors
Determine whether
S={ , } is linearly dependent or independent?
-2
-1
-1/3
2 3
1/3
-2
-2
-1
Halil Aydemir - 19.06.2011
, so linearly independent.
Example: Consider this vectors
and
. Determine whether
S={ , } is linearly dependent or independent?
Thus homogenous system has nontrivial solutions.Therefore , S is lenearly dependent.
Theorem: Let A be an nxn matrix.Thn A is nonsingular iff the columns of A form a linearly
independent set S={ , } ; det(S)=0 this system is linearly dependent.Otherwise
linearly independent.
Definition: A set of vectors S={ ,..., } in a subspace V of is called a basis for V if
S spans V and S is linearly independent.
Note: If system have spans and linearly independent , this system called basis.
Example: The sets
and
are basis Determine the
basis for S and T.
i)
we find constants so S spans .
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Halil Aydemir - 19.06.2011
ii)
so linearly independent.
System S is spans and linearly independent so this system is basis.
i)
we find the constants so S spans
ii)
so linearly independent.
System S is spans and linearly independent so this system is basis.
Theorem: If S={ ,..., } is a basis for subspace V of and is a
linearly independent set of vector in V then
Corallary: S={ ,..., } and are basis for subspace V of then
.
Definition: The dimension of a subspace V of is the number of vectors in a basis for
V.Thus , dimension of is n.
dim( )=n
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Halil Aydemir - 19.06.2011
dim( )=3
Definition: Suppose that S={ ,..., }is basis for a subspace V of and let .The
The vector
is called the coordinate vector of X with respect to S.
Example: Consider the basis
and
for .Let
Find .
for :
for :
Example: Which of the following subset are subspace of ?
i)
. ii)
iii)
i) First is not a subspace because of
but
ii) Second is a subspace because of ;
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Halil Aydemir - 19.06.2011
*
*
iii) Third is not a subspace because of ;
Example: Let
determine if
belongs to spans?
so this system has o solution and
Example: Let
Determine if
belongs to spans.
Therefore ,
Example: Which of the following sets are linearly independent?
i)
ii)
iii)
i)
so this is linearly independent.
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Halil Aydemir - 19.06.2011
ii)
so this system has many solution then linearly dependent.
iii)
so linearly independent.
Example: Consider the subset w of consisting all vectors of the form where a=0.
i) show that w is a subspace of
ii) find a basis for w
iii) find dimension of w
i)
* **Let
***Let
All is true so W is a subspace
ii) * Let
spans W.
**Determine whether S is a linearly independent
so linearly independent.
All is true so S is basis for W.
iii) dim(W)=2 because the number of vectors in S since S has 2 vector.
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Halil Aydemir - 19.06.2011
Example: Consider the subset
of
i)Show that V is a subspace of
ii)Find the basis for V
iii)Find dimension of V
i) *
**Let
so
***
All is true so V is a subspace of
ii) * Take
**
so S is linearly independent.
This equation is both linearly independent and spans.Therefore, S is basis for V.
*** dim(V)=3 b because of 3 vector.
Example: Find the set of vectors spanning the solution space of AX=0 where
Halil Aydemir - 19.06.2011
and so ,
The solution set is
We know that the solution set is the space of AX=0 then;
spans for S.
Note: “AX=0” mean is same with “the null space of A”.
Definition: Let be a set of nonzero vectors in The procedure for
finding a subset of S that is a basis for W=Spans.
#Step1 Form the matrix having as its column vectors.
#Step2 Obtain the reduced row echelon form B of A.
#Step3 The vectors respond’ng to the columns containing the leading entry form a basis for
W=Spans.
Example: Let
Find a subset of S that is a basis for
W=Spans.
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1
1
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Halil Aydemir - 19.06.2011
has 2 leading entry so have 2 basis and dim(B)=2.
Note: “Find a subset of S that is a basis for W=Spans” mean is same with “Find a basis for
the subspace of spanned by S”.
Eigenvalues , Eigenvectors and Diagonalization
Definition: let A be an nxn matrix.The number λ is called an eigenvalues of A if there exists a
nonzero vector X in such that .
A nonzero vector X satisfying is called an eigenvector of A associated with the eigenvalue λ.
This system is a homogenous system.
is called the characteristic polynomial of A.
is called the characterisitic equation of A.
If A is an nxn matrix then is the characteristic polynomial of degree n.
Example: Let
Write characteristic polynomail of A and find eigenvalues
of A.
characteristic polynomial is
eigenvalues are
Theorem: An nxn matrix A is singular iff is an eigenvalue of A.
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Halil Aydemir - 19.06.2011
List of Nonsingular Eigenvalues
* A is nonsingular
*
* rank(A)=n
* A is row equivalent to
* is not an eigenvalue of A
* has only the trivial solution ( has a unique solution ).
Example: The eigenvalue of A are the roots of characteristic polynomial of A.Let
.Find eigenvalues of A and eigenvectors of A.
eigenvalues are
lets find eigenvectors of A;
, so
then
first eigenvector.
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Halil Aydemir - 19.06.2011
, so
then
second eigenvector.
, so
then
third eigenvector.
Definition: A matrix B is similar to A if there is a nonsingular matrix P so that
Some Properties:
* A is similar to A. (
* If B is similar to A then A is similar to B.
* If A is similar to B and b is similar to C then A is similar to C.
Definition: Let A be nxn matrix.A is diagonalizable if A is similar to a diagonal matrix D.
Also we can say that A ca be diagonolized.
Theorem: Similar matrices have the same eigenvalues. The eigenvalues of a diagonal matrix
are the entries on its ain diagonal.
Theorem: An nxn matrix a is diagonalizable iff A has an lineraly independent eigenvectors.
are three linearly independent eigenvectors and A is 3x3.Then A is diagonalizable.
Remark: If A is a diagonizable matrix then where D is a diagonal matrix. The
diagonal elements of D are eigenvalues of A. Moreover , P is matrix whose columns are n
linearly independent eigenvalues of A.
The order of the columns of p determines the order of diagonal elements in D.
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Halil Aydemir - 19.06.2011
Example: Let
Compute the and determine A is
diagonalizable or not.
*
so
** because of the main diagonal
first eigenvector.
second eigenvector.
check that eigenvectors are linearly independent , if it is , A is diagonalizable.
This system is unique solution and constants are zero so linearly independent.Then A is
diagonalizable.
Example: Let
if possible ; find matrix P that diagonlizes A and determine
so
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Halil Aydemir - 19.06.2011
then is the eigenvector assicated with .
then is the eigenvector assicated with .
There don’t exist three linearly independent eigenvector of A.Therefore , A is not
diagonalizable.
.
Example: Let
if possible ; find matrix P that diagonlizes A and determine
.
so
, ,
then
is the eigenvector assicated with .
, ,
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-1/2
-1
2
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1
4
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1
Halil Aydemir - 19.06.2011
then
is the eigenvector assicated with .
, ,
then
is the eigenvector assicated with
Therefore ; are linearly independent eigenvector since the rrots of characteristic
polynomial of A distrinct.A is diagonalizable.
Note: For writing , put the each value on the main diagonal.
Definition: Let A be an nxn matrix. Then the characteristic polynomial of A is
=
. The characteristic equation is
This equation has at most n distinct roots. A
has almost n distinct eiqenvalues for so
Example: Show that A and have the same eigenvalues.
Example: Let A be an nxn matrix ;
a)Show that |A| is the product of all the roots of the characteristic polynomial of A.
b)Show that A is singular if and only if is an eigenvalues of A.
a)
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Halil Aydemir - 19.06.2011
b) A is singular i.e. such that i.e. 0 is an
eigenvalues of A.
Coyley – Hamilton Theorem
If is the characteristic polynomial of A then
Example: Verify Coyley-Hamilton theorem for the following matrix
.
so
correct.
Example: Use the Coyley-Hamilton theorem to compute the inverse of
.
then
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Halil Aydemir - 19.06.2011
Note: .This mean for using Coyley-Hamilton
thorem , determinant must be different than zero.