Matrix Calculus Flashcards
Determining a minor in a matrix
Remove the elements in the target’s row and column, including the target itself
Calculate the determinant of the new matrix
Matrix determinant
Only applies to a square matrix
Difference ‘meaning subtracting’ of the individual products of the numbers listed diagonally from the top-left corner. When you run out of space, go to the next row and start at the first column
Starting position in a matrix
i=1, j=1
Start in the upper-left (like reading a book)
Dimensions of a matrix, and propper notation for element identity
Rows-by-columns
Referring to an individual item in a matrix
(Matrix-symbol-title)ij
Where i is rows from top and j is columns from left
Matrix addition/subtraction
Add/subtract each element to the element of that position within the other matrix
Scalar multiplication of matrices
Multiply each element within the matrix by the scalar-constant
Transposing matrices
Row 1 items fill column 1 of the new matrix and so on for each row
Writen as [matrix]^T
Size of a product matrix in Matrix Multiplication
Rows1 by columns2
Matrix multiplication
Sum of the products of the next element in matrix1 row and matrix2 column corresponding to the element location in the product matrix
Row addition
Adding an entire row of one matrix to the entire row of another matrix
Row multiplication
Multiplying an entire row of a matrix by a scalar (constant number)
Row switching
Switching the position of two rows in a matrix
Sub-matrices
The matrix that excludes all the elements within the row or column of a given position in the original matrix
Linear equations from matrices
When the formula for the value of the element in a given position is writen out, it takes the form of a linear equation (like in circuit analysis)
Square matrix
A matrix with the same number of rows as columns
Diagonal matrices
A square-matrix in which all the elements outside of an imaginary diagonal line in the matrix are equal to zero
Triangular matrix
A square-matrix in which all the elements outside of an imaginary triangle within that matrix have a value of zero
Identity matrix
A square-matrix in which all the elements on the main diagonal have a value of 1
Main diagonal within a matrix
Starts at initial position (11/upper left) and runs down the the lower right corner in a square matrix
Symmetric Matrices
A matrix which is equal to its own transpose
Skew-symmetric matrix
A matrix which is equal to its own NEGATIVE transpose
meaning the scalar-product of the transpose and -1
Geometric shapes from 2x2 matrices
Assume that the point (0,0) is a vertecy of the figure
Both columns are their own point (x,y) read top-down
And the sum of the elements in a row from the top down as (x,y)
The enclosed area is the geometric figure
Horizontal shear transformation for geometric figures by a 2x2 matrix
Element 12 increases
Horizontal flip transformation for geometric figures by a 2x2 matrix
Element 11 is made negative
Verticle flip transformation for geometric figures by a 2x2 matrix
Element 22 is made negative
Squeeze flip transformation for geometric figures by a 2x2 matrix
Element 11 becomes the reciprocal of the fractional element 22
Scaling transformation for geometric figures by a 2x2 matrix
Element 11 and element 22 are multiplied by the same scalir-factor
Rotational ransformation for geometric figures by a 2x2 matrix
Elements 11 and 22 are multiplied by cos(angle) While sin(angle) is added to element 21 and subtracted from element 12
Inverse matrix
A matrix writen as A^-1
The product of this matrix and another would be the same as dividing that matrix by ‘A’, which you don’t get to do with matrices
Matrix functions
A function into which the matrix is an input to be changed as it into an output
Can include transposing, transformation, scalar, multiplication, addition, or subtraction
Orthoganal Matrix
A matrix for which the transpose is equal to the inverse
Matrix trace
Sum of its diagonal elements
Writen as tr(A), where ‘A’ is the matrix
Gradient Matrix
A matrix formed from the differentials that describe the vector gredient
Usually 1x3
Matrices of a vector
Take the form 3x1, go down x, y, z
Matrices of the derivative of a scalar vector (tangent vector)
If x is the scalar, each element is
d(element)/dx
Orthogonal projection of a vector matrix
A=1/||u||^2 [(ux)^2, (ux)(uy)]
[(ux)(uy), (uy)^2 ]
Reflecting a vector/matrix about a line that goes through the origin
A=1/||u||^2 [(ux)^2-(uy)^2, 2(ux)(uy)]
[2(ux)(uy), (uy)^2-(ux)^2]
Write it out
Hessian Matrix
A square matrix of second order partial derivatives of a function
Eigenvector of matrix transformation
The vector within that plane the direction and position of which remains completely unchanged by the transformation, but about which the transformation occurs
Eigenfunction
A transformation of a matrix for which a given vector in that plane, by definition the eigenvector, remains completely unchanged
Use for transformations of a matrix
Used to describe a change in 2D or 3D space
1) Expansion through an object
2) Vibration through a surface
3) Proximity and direction from one electron to another
4) Propagation of light through an interface
See laplase transforms
Normal mode of a matrix transformation
The pattern (frequency and rmagnetude).by which a set of transformations repeat themselves within a given system
Minor of a Matrix element
The determinant of the matrix that excludes the elements in the rows or columns of that element
