matrices and matrix methods Flashcards
How would a 2-dimensional image be typically be represented?
256 x 256 matrix
What is a matrix?
A set of real or complex numbers arranged in rows + columns to form a rectangular array
What is m x n matrix?
A matrix having m rows and n columns
What does row matrix consist of?
1 row only
What do column matrix consist of?
1 column only
What does square matrix have?
Equal number of rows and columns
What do each element in a matrix have?
Its own ‘address’ or ‘location’ defined by a systems of double suffixes
What do the first and second suffix indicate?
First = row Second = column
How can a 3D matrix be written as?
(aijk)
How can a vector be represented as?
A matrix of order 1 x n
How can matrices be added or subtracted?
2 matrices must be of the same order
What is the application of addition and substraction of matrices?
Imaging application
Adding multiple averages to increase SNR
How can matrixes be multiplied?
By single number (i.e. scalar)
Each individual element is multiplied by that factor
What is the application of multiplication of matrices?
Changing image display brightness
What is equation for multiplication of matrices?
k(aij) = (kaij)
When is matrix multiplication only possible?
When no. of column in first is equal to the number of rows in second
a.b = c
When are transpose of a matrix formed?
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
What is a square matrix a order of?
m x m
What is a square matrix?
symmetric
symmetric about leading diagonal
if aij=aji
What is a diagonal matrix?
A square matrix with all elements 0 except those on the leading diagonal
What is unit matrix?
A diagonal matrix in which elements along the leading diagonal are unity
What is the application of special matrices?
Imaging application:
symmetric and diagonal matrices feature in DTI theory
What is the unit matrix denoted by and what it is its role?
Denoted by I
I has the same function as the number 1 in standard scalar algebra
What is the consequence of multiplying matrix by identity matrix I?
Leaves it unchanged
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)
What do determinant of matrix provide information on?
Useful in matrix representation of linear equations
Useful when matrix represents a geometric transformation
How is determinat of 2 x 2 matrix written symbollically as?
det A or magnitude of A
[A B C D] =
ad-bc
What is the determinant of a 3x3 matrix?
(aei+bfg+cdh)-(ceg+afh+bdi)
What is the determinant?
Just a number
It gives useful information about the matrix
It is needed to compute the inverse of the matrix
What is matrix division?
Inverse of matrix
Only applies to square matrices
What is the important application of a matrix division?
Image transformation
Image analysis
Performed in software
How is the inverse of a 2x2 matrix determined by?
Rearranging the matrix elements and dividing by the determinant
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
