Matrices #1

What is a row and column

row ……..

.

.

.column

.

.

How to do following operations with vectors

1, scalar multiplication

2,addition

3,subtraction

1, multiplying all the vector elements with multiplicator

2,adding corresponding elements of the vectors

3, subtracting corresponding elements of the vectors

How do we add/subtract vectors of different levels

the result is not defined

What is inner/scalar product of two vectors

v´w = ∑ v(i) * w(I)

result is one number and the rule is basically as with matrix multiplication

What does

1, transpose vector

2, orthogonality

mean

1, transpose vector v´is in a row instead of column

2, orthogonality of 2 vectors= their scalar product is 0

What is dimension of this Matrix

( 1 2 3 )

( 4 5 6)

2×3 Matrix

What is a12 & a21 of this Matrix

( 1 2 3 )

( 4 5 6)

a12 = 2

a21= 4

( 1 2 3 )

( 4 5 6)

How do we multiplicate matrices

next matrix element = scalar product of row of first matrix with the column of the second matrix

What is

1,dimensionality

2,conformity

1, dimensionality

if A = n*m & B = m*p then after multiplication the dimensions are gonna be n*p

2, conformity

if A = n*m & B=m*p then m on right (first matrix) and on left (second matrix) mean we can multiply them

What is associative law when multiplying matrices

associative law .. it doesnt matter what we multiply first if ranking stays

(A*B)*C = A*(B*C)

What is distributive law

(A+B)C = AC + AB

A(B+C) = AB + AC

just as ordinary numbers just the ranking has to stay the same

How to calculate power of matrix

A^n = A*A*A….*A

except A^0 = I (identity matrix)

What is Kronecker Product

written A⊗B

A⊗B=

(a11*B, a12*B)

(a21*B, a22*B)

What are the new dimension of Kronecker Product

A=m x n & B=p x q

then A⊗B = m*p x n*q

What is Idempotent matrix

idempodent matrix satsfies

A² = A

What is the trace of a matrix

the sum of all diagonal elements

tr(A) = ∑a(i,i)

What is inverse matrix (2)

1,A^-1 » if AxA^-1 = I (identity matrix)

2, if inverse matrix exist = regular/non-singular

if no inverse matrix exist = singular

What is

1, (AB) ^-1

2, (A´)^-1

1, (AB) ^-1 = B^-1 * A^-1

2, (A´)^-1 = (A^-1)´

What does determinant says about inverse matrix

if det(A)=0 it is singular

if det(A) ≠ 0 it has a inverse matrix