Download Linear Algebra (wi1403lr)

Linear Algebra (wi1403lr)
Lecture no.6
EWI / DIAM / Numerical Analysis group
Matthias Möller
02/05/2014
M. Möller (EWI/NA group)
LA (wi1403lr)
02/05/2014
1 / 14
Review of lecture no.5
2.1 Matrix operations
• sums and scalar multiples of matrices
• matrix-matrix multiplication
• powers of a matrix
• transpose of a matrix
M. Möller (EWI/NA group)
LA (wi1403lr)
02/05/2014
2 / 14
Learning objectives of lecture no.6
You will learn
• to compute the inverse of a quadratic matrix (if it exists)
• to decide whether a matrix is invertible
• to compute the determinant of a quadratic matrix
• to use the concept of invertible matrices to compute the inverse
transformation from the codomain back to the domain
M. Möller (EWI/NA group)
LA (wi1403lr)
02/05/2014
3 / 14
Multiplicative inverse
Let a be an arbitrary scalar.
The formal way to consider the division operation 1/a is as follows:
1
Definition: We seek the multiplicative inverse of a such that
a−1 · a = 1
and a · a−1 = 1
2
Existence: The notation for it is a−1 and it exists if a 6= 0.
3
Uniqueness: The multiplicative inverse is unique.
M. Möller (EWI/NA group)
LA (wi1403lr)
02/05/2014
4 / 14
Inverse of a matrix
Let A be an ’arbitrary’ n × n matrix.
1
Definition:
2
Existence:
3
Uniqueness:
M. Möller (EWI/NA group)
LA (wi1403lr)
02/05/2014
5 / 14
Inverse of a matrix
Let A be an ’arbitrary’ n × n matrix.
1
Definition: We say that A is invertible if there exists an n × n
matrix A−1 such that
A−1 A = I
and AA−1 = I
with I being the n × n identity matrix. If such inverse matrix
does not exist then we call A a singular matrix.
2
Existence:
3
Uniqueness:
M. Möller (EWI/NA group)
LA (wi1403lr)
02/05/2014
5 / 14
Inverse of a matrix
Let A be an ’arbitrary’ n × n matrix.
1
Definition: We say that A is invertible if there exists an n × n
matrix A−1 such that
A−1 A = I
and AA−1 = I
with I being the n × n identity matrix. If such inverse matrix
does not exist then we call A a singular matrix.
2
Existence:
3
Uniqueness: The inverse matrix is unique.
If there was another inverse Ã−1 then
Ã−1 = Ã−1 I = Ã−1 (AA−1 ) = (Ã−1 A)A−1 = IA−1 = A−1
M. Möller (EWI/NA group)
LA (wi1403lr)
02/05/2014
5 / 14
Inverse of a matrix
Let A be an ’arbitrary’ n × n matrix.
1
Definition: We say that A is invertible if there exists an n × n
matrix A−1 such that
A−1 A = I
and AA−1 = I
with I being the n × n identity matrix. If such inverse matrix
does not exist then we call A a singular matrix.
2
Existence: ???
3
Uniqueness: The inverse matrix is unique.
If there was another inverse Ã−1 then
Ã−1 = Ã−1 I = Ã−1 (AA−1 ) = (Ã−1 A)A−1 = IA−1 = A−1
M. Möller (EWI/NA group)
LA (wi1403lr)
02/05/2014
5 / 14
Computing the inverse of a 2 × 2 matrix
a b
Let A =
. If ad − bc 6= 0, then A is invertible and
c d
−1
A
1
d −b
=
ad − bc −c a
If ad − bc = 0, then A is not invertible.
This quantity is termed the determinant of matrix A
det A = ad − bc
Decide if the inverse of matrix A =
1 2
exists and compute it if
5 12
this is possible.
M. Möller (EWI/NA group)
LA (wi1403lr)
02/05/2014
6 / 14
Take home lesson
a b
Let A =
be an ’arbitrary’ 2 × 2 matrix.
c d
1
Definition: The inverse matrix is defined as
1
d −b
−1
A =
ad − bc −c a
provided that ad − bc 6= 0.
2
Existence: The inverse matrix exists if and only if
det A = ad − bc 6= 0
3
Uniqueness: The inverse matrix is unique.
M. Möller (EWI/NA group)
LA (wi1403lr)
02/05/2014
7 / 14
Solution of linear equations
If the n × n matrix A is invertible, then for each vector b in Rn , the
equation Ax = b has the unique solution x = A−1 b.
1 2
6
−1
−1
Let A =
with inverse matrix A =
.
5 12
−2.5 0.5
Solve the linear equations
−1
Ax =
3
M. Möller (EWI/NA group)
1
Ax =
−5
LA (wi1403lr)
2
Ax =
6
02/05/2014
8 / 14
Solution of linear equations
If the n × n matrix A is invertible, then for each vector b in Rn , the
equation Ax = b has the unique solution x = A−1 b.
1 2
6
−1
−1
Let A =
with inverse matrix A =
.
5 12
−2.5 0.5
Solve the linear equations
−1
Ax =
3
1
Ax =
−5
2
Ax =
6
There must be a less tedious way to do this!
M. Möller (EWI/NA group)
LA (wi1403lr)
02/05/2014
8 / 14
Draft of a solution idea
Write your problem(s) into augmented form
A b1 b2 b3
and transform your matrix A into the 2 × 2 identity matrix by
applying only elementary row operations:
1
Add/subtract the multiple of one row to another row.
2
Scale one row by a scalar.
3
Swap the position of two rows.
M. Möller (EWI/NA group)
LA (wi1403lr)
02/05/2014
9 / 14
Draft of a solution idea
Write your problem(s) into augmented form
A b1 b2 b3
and transform your matrix A into the 2 × 2 identity matrix by
applying only elementary row operations:
1
Add/subtract the multiple of one row to another row.
2
Scale one row by a scalar.
3
Swap the position of two rows.
We found a simple algorithm to solve a sequence of linear equations
with the same matrix A and arbitrarily many right-hand sides bi .
M. Möller (EWI/NA group)
LA (wi1403lr)
02/05/2014
9 / 14
Computing the inverse of a matrix
1 2
1
0
Let A =
. Solve the linear equations Ax =
and Ax =
5 12
0
1
by writing it in augmented form and applying elementary operations.
M. Möller (EWI/NA group)
LA (wi1403lr)
02/05/2014
10 / 14
Computing the inverse of a matrix
1 2
1
0
Let A =
. Solve the linear equations Ax =
and Ax =
5 12
0
1
by writing it in augmented form and applying elementary operations.
1 2 | 1 0
1 0 |
6
−1
∼ ··· ∼
5 12 | 0 1
0 12 | −2.5 0.5
M. Möller (EWI/NA group)
LA (wi1403lr)
02/05/2014
10 / 14
Computing the inverse of a matrix
1 2
1
0
Let A =
. Solve the linear equations Ax =
and Ax =
5 12
0
1
by writing it in augmented form and applying elementary operations.
1 2 | 1 0
1 0 |
6
−1
∼ ··· ∼
5 12 | 0 1
0 12 | −2.5 0.5
An n × n matrix A is invertible if and only if A is row equivalent to I ,
and in this case, any sequence of elementary row operations that
reduces A to I also transforms I into A−1 .
M. Möller (EWI/NA group)
LA (wi1403lr)
02/05/2014
10 / 14
Algorithm for finding A−1
1
Write down the augmented matrix
A I
2
Row reduce the augmented matrix to
I A−1
by applying only elementary row operations.
If the algorithm breaks down before A has been transformed (i.e.
row reduced) into I , then A does not have an inverse.
M. Möller (EWI/NA group)
LA (wi1403lr)
02/05/2014
11 / 14
Properties of the inverse of a matrix
• If A is an invertible matrix, then A−1 is also invertible and
(A−1 )−1 = A
• If A and B are n × n invertible matrices, then so is AB, and
(AB)−1 = B −1 A−1
• If A is an invertible matrix, then so is AT , and
(AT )−1 = (A−1 )T
Proofs of each statement can be found in the book.
M. Möller (EWI/NA group)
LA (wi1403lr)
02/05/2014
12 / 14
The invertible matrix theorem
Let A be a square n × n matrix. Then the following statements are
are either all true all false, that is, they are equivalent.
a. A is an invertible matrix.
...
l. AT is an invertible matrix.
Proof (by showing equivalence of the two statements)
M. Möller (EWI/NA group)
LA (wi1403lr)
02/05/2014
13 / 14
The invertible matrix theorem
Let A be a square n × n matrix. Then the following statements are
are either all true all false, that is, they are equivalent.
a. A is an invertible matrix.
...
l. AT is an invertible matrix.
Proof (by showing equivalence of the two statements)
• (a) ⇒ (l): By the third property on the previous slide.
M. Möller (EWI/NA group)
LA (wi1403lr)
02/05/2014
13 / 14
The invertible matrix theorem
Let A be a square n × n matrix. Then the following statements are
are either all true all false, that is, they are equivalent.
a. A is an invertible matrix.
...
l. AT is an invertible matrix.
Proof (by showing equivalence of the two statements)
• (a) ⇒ (l): By the third property on the previous slide.
• (l) ⇒ (a): By the third property on the previous slide with A and
AT interchanged.
M. Möller (EWI/NA group)
LA (wi1403lr)
02/05/2014
13 / 14
The invertible matrix theorem
Let A be a square n × n matrix. Then the following statements are
are either all true all false, that is, they are equivalent.
a. A is an invertible matrix.
...
l. AT is an invertible matrix.
Proof (by showing equivalence of the two statements)
• (a) ⇒ (l): By the third property on the previous slide.
• (l) ⇒ (a): By the third property on the previous slide with A and
AT interchanged.
• This implication in both directions proves that both statements
are equivalen, i.e. (a) ⇔ (l).
M. Möller (EWI/NA group)
LA (wi1403lr)
02/05/2014
13 / 14
The invertible matrix theorem
Let A be a square n × n matrix. Then the following statements are
are either all true all false, that is, they are equivalent.
a. A is an invertible matrix.
j. There is an n × n matrix C such that CA = I .
d. The equation Ax = 0 has only the trivial solution.
c. A has n pivot positions.
b. A is row equivalent to the n × n identity matrix.
Proof (by establishing a circle of implications)
(a) ⇒ (j) ⇒ (d) ⇒ (c) ⇒ (b) ⇒ (a)
M. Möller (EWI/NA group)
LA (wi1403lr)
02/05/2014
14 / 14