a very brief linear algebra review

8/13/2019 A Very Brief Linear Algebra Review

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A VERY BRIEF LINEAR ALGEBRA REVIEW

Matrices. ExampleA =

2 11 3

A is a matrix with 2 rows and 2 columns i.e a 2 2 matrix.A matrix with the same number of rows and columns is called a square matrix.Example of a 3 3 matrix

B =

3 1 71 2 0

0 1 5

Example of a 3 2 matrix

C = 2 09 101 14

Vectors. Matrices with 1 row are called row vectors and matrices with 1 columnare called column vectors.

A =

21

B =

32

1

are column vectors.

C = 2 1 D = 3 2 1are row vectors.

Addition Of Two Matrices.2 11 3

+

1 01 2

=

3 10 5

Multiplication Of Two Matrices. 2 1 31 1 2

3 1 1

10

2

=

21 + 10 + 3 2(1)1 + 10 + 22

31 + 10 + 1 2

=

83

5

2 1 31 0 12 1 0

1 1 01 2

1

2

3 0 1

=

21 + 11 + 3 3 2(1) + 12 + 30 20 + 1 12 + 3111 + 01 + 1 3 1(1) + 02 + 10 10 + 0

1

2 + 1121 + 11 + 0 3 2(1) + 12 + 00 20 + 1 1

2+ 01

Note : Multiplication AB can be done only if the number of columns of A is the

same as the number of rows of B.

Transpose Of A Matrix. Let A=

2 1 10 1 21 0 1

then At =

2 0 11 1 01 2 1

1


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Properties of Transpose.

(AB)t

= BtAt

(A + B)t

= At + Bt

Complex Numbers. Any number z = a +bi where a and b are real numbers iscalled a complex number.

Complex numbers can be used in matrices.

A =

1 +i 23i 1

2+ 2i

=

1 23 1

2

+

1 01 2

i

Complex Conjugation. Ifz = a +bi where a and b are real numbers, then thecomplex conjugate

ofz

isz

=a

bi.

Adjoint of a Matrix. The adjoint of a matrix A denoted as A is At

For the matrix A given above,

A = At =

1i 3 +i

2 12 2i

Properties of Adjoint.

(AB) = BA

(A + B) = A + B

Symmetric Matrices. A matrix B is symmetric if B = Bt

Example: 3 1 21 0 52 5 2

is symmetric.

Hermitian (Self-adjoint) Matrices. A matrix B is Hermitian if B = BExample

2 1 +i 12 + 2i1i 3 51

2 2i 5 4

is Hermitian.

Vector Cross Product. ab

c

xy

z

=

i j ka b c

x y z

= (bzcy)i (azxc)j + (aybx)k

=

bzcyaz+xc

aybx

.


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Determinant.

deta b

c d

= adbc=a b

c d

det

a b cd e f

g h k

= a e fh k

bd fg k

+cd eg h

Dot Product. Let v=

ab

c

and w =

xy

z

then

vw = vtw=

a b c xy

z

= ax+by+cz

Hermitian Dot Product. Let v=

1 +i1

i

and w=

21i

3

.

Then

v, w= v w= (1i)2 + 1(1i) + (i)3.

Identity Matrix. The matrix I is defined as the matrix whose entries are aijwhere

aij =

1 ifi = j

0 ifi =j.

The matrix I is called the unit or identity matrix.

Inverse Of A Matrix. Let A be a square (n x n) matrix. If detA= 0 there is aninverse A1 such that A1A= AA1 =I .

If

A=

a11 a12a21 a22

then

A1 = 1

a22 a12a21 a11

where = a11a22a12a21 is the determinant ofA.


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Eigenvalue and Eigenvectors. is an eigenvalue of a matrix A correspondingto an eigenvector V = 0 ifAV =V.

Example 1 2 11 2

11

= 3

11

The eigenvalue is 3, the eigenvector is (1, 1)t.

Example 2 0 11 0

1i

=

i

1

= i

1i

The eigenvalue is i, the eigenvector is (1,i)t.

Example 3 0 11 0

1i

=

i1

= i

1i

The eigenvalue is i , the eigenvector is (1, i)t.

a very brief linear algebra review

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