matrices and matrix methods
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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