matrices and matrix methods Flashcards

1
Q

How would a 2-dimensional image be typically be represented?

A

256 x 256 matrix

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2
Q

What is a matrix?

A

A set of real or complex numbers arranged in rows + columns to form a rectangular array

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3
Q

What is m x n matrix?

A

A matrix having m rows and n columns

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4
Q

What does row matrix consist of?

A

1 row only

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5
Q

What do column matrix consist of?

A

1 column only

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6
Q

What does square matrix have?

A

Equal number of rows and columns

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7
Q

What do each element in a matrix have?

A

Its own ‘address’ or ‘location’ defined by a systems of double suffixes

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8
Q

What do the first and second suffix indicate?

A
First = row
Second = column
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9
Q

How can a 3D matrix be written as?

A

(aijk)

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10
Q

How can a vector be represented as?

A

A matrix of order 1 x n

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11
Q

How can matrices be added or subtracted?

A

2 matrices must be of the same order

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12
Q

What is the application of addition and substraction of matrices?

A

Imaging application

Adding multiple averages to increase SNR

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13
Q

How can matrixes be multiplied?

A

By single number (i.e. scalar)

Each individual element is multiplied by that factor

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14
Q

What is the application of multiplication of matrices?

A

Changing image display brightness

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15
Q

What is equation for multiplication of matrices?

A

k(aij) = (kaij)

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16
Q

When is matrix multiplication only possible?

A

When no. of column in first is equal to the number of rows in second

a.b = c

17
Q

When are transpose of a matrix formed?

A

When the rows and columns of matrix are interchanged

  • The first row becomes the column
  • The second row becomes the second column
  • The third row becomes the third column
18
Q

What is a square matrix a order of?

19
Q

What is a square matrix?

A

symmetric
symmetric about leading diagonal

if aij=aji

20
Q

What is a diagonal matrix?

A

A square matrix with all elements 0 except those on the leading diagonal

21
Q

What is unit matrix?

A

A diagonal matrix in which elements along the leading diagonal are unity

22
Q

What is the application of special matrices?

A

Imaging application:

symmetric and diagonal matrices feature in DTI theory

23
Q

What is the unit matrix denoted by and what it is its role?

A

Denoted by I

I has the same function as the number 1 in standard scalar algebra

24
Q

What is the consequence of multiplying matrix by identity matrix I?

A

Leaves it unchanged

25
What is determinant of a matrix?
A value associated with square matrix Computed from the entries of the matrix by a specific arithmetic expression (or using computer function)
26
What do determinant of matrix provide information on?
Useful in matrix representation of linear equations | Useful when matrix represents a geometric transformation
27
How is determinat of 2 x 2 matrix written symbollically as?
det A or magnitude of A [A B C D] = ad-bc
28
What is the determinant of a 3x3 matrix?
(aei+bfg+cdh)-(ceg+afh+bdi)
29
What is the determinant?
Just a number It gives useful information about the matrix It is needed to compute the inverse of the matrix
30
What is matrix division?
Inverse of matrix | Only applies to square matrices
31
What is the important application of a matrix division?
Image transformation Image analysis Performed in software
32
How is the inverse of a 2x2 matrix determined by?
Rearranging the matrix elements and dividing by the determinant
33
Inverse of 2x2 matrix?
The elements on the leading diagonal are swapped, the remaining elements are multiplied by -1, and the resulting matrix divided by the determinant of A