Algebra of Matrices

Question 1

Identity Matrix

Answer 1

A matrix that has only 1’s in it’s diagnol entries and zero’s elsewhere.

Question 2

scalar matrix

Answer 2

A matrix that has scalar non zero entries as it’s diagnol entries and zero’s elsewhere.

Question 3

Matrix

Answer 3

A set of numbers arranged in row column form to form a matrix.

Question 4

Addition or substraction of a matrix

Answer 4

A matrix can only be substracted or added if both matrices have same dimension i.e no of rows and columns are equal

Question 5

Matrix Multiplication

Answer 5

A matrix can only be multiplied if the:

no of columns of first matrix = no of rows of second column.

1st row 1st column
1st row second column
1st row third column and…
2nd row 1st column
2nd row second column
2nd row third column…

To determine the dimension of the resulting matrix can be determined by rows of 1st matrix and column of 2nd matrix.

Question 6

Matrix Transpose

Answer 6

Interchanging rows into columns

Question 7

Nonsingular (Invertible matrix)

Answer 7

A matrix that has non zero determinant is called non singualr matrix and only a non singular matrix is invertible.

An invertible matrix is a square matrix that has an inverse, which is another square matrix that satisfies two conditions:

1.When the inverse is multiplied by the original matrix, the result is the identity matrix of the same size as the matrices being multiplied. In other words, AB = BA = I, where A is the original matrix, B is its inverse, and I is the identity matrix.

2.The inverse is unique, meaning that there is only one matrix B that satisfies the first condition for a given matrix A.

Question 8

Diagnol Matrix

Answer 8

A diagnol matrix is a square matrix in which all non diagnol entries are zero and the diagnol entries can be any real, complex numbers or even zero.

Question 9

Triangular Matrix

Answer 9

A matrix is upper triangular if the entries below the main diagnol are all non zero.
A matrix is called lower triangular if the entries above the main diagnol are all non zero.

Question 10

Symmetric Matrix

Answer 10

A matrix is called symmetric if the A transpose is equal to A.
Example: A = [ 3 4 5 ]
[ 4 6 7 ]
[ 5 7 9 ]

Question 11

Skew symmetric matrix

Answer 11

A matrix is called skew symmetric if A transpose is equal to -A e.g multiplying a with minus.

Question 12

Orthongonal Matrix

Answer 12

A matrix is orthongonal if A multiply with A transpose gives us an identity matrix.

Question 13

Block matrix

Answer 13

A block matrix is a matrix that is composed of smaller matrices called blocks. the matrix blocks meaning the submatrices themselves are not square matrices instead when combined they become square.

Question 14

Block Diagnol Matrices

Answer 14

A block diagonal matrix is a matrix that is composed of square submatrices along the diagonal, with all other entries outside of the diagonal being zero. it can as well be upper triangular or lower triangular.

Question 15

Block Square Matrices

Answer 15

a block square matrix is a matrix in which along the diagnol they should be square either 1 by 1 2 by 2 3 by3 and outside the diagnol it does not matter what dimension they have

Question 16

Complex Conjugate

Answer 16

complex conjugate is only with complex numbers and it means changing the symbols of imaginary numbers e.g. So, the complex conjugate of 1+2i is 1-2i, the complex conjugate of 3-4i is 3+4i, and the complex conjugate of 6+7i is 6-7i.

A^H means CONJUGATE TRANSPOSE OF A. meaning we will take its transpose and then conjugate

Question 17

Complex normal matrix

Answer 17

A complex matrix A is said to be normal if it commutes (two operations commute, it means that the order in which we perform them does not matter. In other words, we can perform the two operations in any order, and the final result will be the same) with A^H—that is, if
AA^H = A^HA.

Question 18

Normal Matrix

Answer 18

A real matrix A is normal if it commutes with its transpose A^T—that is, if AA^T = A^TA

Question 19

Hermitian and Skew Hermitian Matrix

Answer 19

A matrix is Hermitian if:
A^H= A
A matrix is Skew Hermitian if:
A^H=minus A
:

Question 20

what are not done in chapter 1

Answer 20

Orthogonal Matrix
Proving Theorem
Unitary matrix
Definitions
solving fractions

Question 21

Unitary Matrix

Answer 21

It should be Square and Complex. AA^H=I.