Unit 1 - Introduction to Matrices

Question 1

Matrix definition

Answer 1

It’s a rectangular grouping of numbers or symbols arranged in rows and columns.

To understand the meaning of a matrix, row and columns has to be described in advance.

Question 2

m x n, whic is row which is column?

Answer 2

m = row
n = columns

Question 3

labeling

Answer 3

the matrix is labeled with a capital letter: A
the elements are labeled with lowercase letters a i,j

Question 4

Transposition definition

Answer 4

The mathematical permutation involves the exchange of two elements.
In matrix means to swap row and columns with each other.

Question 5

Column vector

Answer 5

m x 1, a matrix with only one column.

Question 6

Row vector

Answer 6

1 x n, a matrix with only one row.

Question 7

Zero matrix

Answer 7

A matrix whose elements are all zero.

Question 8

Main diagonal

Answer 8

Is the diagonal whose elements are formed with the same row and column index.

Question 9

Diagonal matrix

Answer 9

D = (d i,j) is a matrix whose elements outside the main diagonal are zero.

Question 10

Upper triangular matrix

Answer 10

The elements below the main diagonal are zero.
U i,j = 0 foralli>j

Question 11

Lower triangular matrix

Answer 11

The elements above the main diagonal are zero.
L i,j = 0 foralli<j

Question 12

Band matrix

Answer 12

Only have non-zero elements in the vicinity of the main diagonal.

Question 13

Square matrix

Answer 13

A matrix that have the same number of rows and columns.
m x n, m=n, rows = columns

Question 14

Addiction and subtraction of matrices can be done only when?

Answer 14

Only matrices of the same type m x n, which means matrix A and matrix B must have the same number of rows and columns.

Question 15

Matrix Addition laws

Answer 15

Cumulative law: A+B = B+A

Associative law: A+(B+C) = (A+B)+C

The zero matrix is the neutral element: A+0 = 0+A =A

A-A = 0

Question 16

When a square matrix is symmetric?

Answer 16

If A = A^T

Every square diagonal matrix is a symmetrical matrix

Every identity matrix is a symmetrical matrix

Question 17

Skew-symmetric matrix

Answer 17

A = -A^T

Question 18

What is a scalar?

Answer 18

A real number that can be multiplied to a matrix.

Question 19

Law of calculations of scalar

Answer 19

Associative law: (λ⋅μ)⋅A=λ⋅(μ⋅A)

Distributive law: (λ+μ)⋅A=λ⋅A+μ⋅A ; λ⋅(A+B)=λ⋅A+λ⋅B

with transposing: (A+B)^T=A^T+B ^T
; (λ-A)^T=λ-A^T

Question 20

The dot product vectors can be done only?

Answer 20

Only for vectors that have the same number of elements.

Question 20

Decomposition theorem for square matrix

Answer 20

A = B + C

B = 1/2*(A+A^T)

C = 1/2*(A-A^T)

Question 21

Calculation rules for vectors with the same number of elements and for any scalar λ∈R

Answer 21

.