Matrix Calculus Flashcards Preview

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Flashcards in Matrix Calculus Deck (44):
1

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

2

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

3

Starting position in a matrix
(i=1, j=1)

Start in the upper-left (like reading a book)

4

Dimensions of a matrix, and propper notation for element identity

Rows-by-columns

5

Referring to an individual item in a matrix

(Matrix-symbol-title)ij
Where i is rows from top and j is columns from left

6

Matrix addition/subtraction

Add/subtract each element to the element of that position within the other matrix

7

Scalar multiplication of matrices

Multiply each element within the matrix by the scalar-constant

8

Transposing matrices

Row 1 items fill column 1 of the new matrix and so on for each row

Writen as [matrix]^T

9

Size of a product matrix in Matrix Multiplication

Rows1 by columns2

10

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

11

Row addition

Adding an entire row of one matrix to the entire row of another matrix

12

Row multiplication

Multiplying an entire row of a matrix by a scalar (constant number)

13

Row switching

Switching the position of two rows in a matrix

14

Sub-matrices

The matrix that excludes all the elements within the row or column of a given position in the original matrix

15

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)

16

Square matrix

A matrix with the same number of rows as columns

17

Diagonal matrices

A square-matrix in which all the elements outside of an imaginary diagonal line in the matrix are equal to zero

18

Triangular matrix

A square-matrix in which all the elements outside of an imaginary triangle within that matrix have a value of zero

19

Identity matrix

A square-matrix in which all the elements on the main diagonal have a value of 1

20

Main diagonal within a matrix

Starts at initial position (11/upper left) and runs down the the lower right corner in a square matrix

21

Symmetric Matrices

A matrix which is equal to its own transpose

22

Skew-symmetric matrix

A matrix which is equal to its own NEGATIVE transpose
meaning the scalar-product of the transpose and -1

23

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

24

Horizontal shear transformation for geometric figures by a 2x2 matrix

Element 12 increases

25

Horizontal flip transformation for geometric figures by a 2x2 matrix

Element 11 is made negative

26

Verticle flip transformation for geometric figures by a 2x2 matrix

Element 22 is made negative

27

Squeeze flip transformation for geometric figures by a 2x2 matrix

Element 11 becomes the reciprocal of the fractional element 22

28

Scaling transformation for geometric figures by a 2x2 matrix

Element 11 and element 22 are multiplied by the same scalir-factor

29

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

30

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

31

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

32

Orthoganal Matrix

A matrix for which the transpose is equal to the inverse

33

Matrix trace

Sum of its diagonal elements
Writen as tr(A), where 'A' is the matrix

34

Gradient Matrix

A matrix formed from the differentials that describe the vector gredient
Usually 1x3

35

Matrices of a vector

Take the form 3x1, go down x, y, z

36

Matrices of the derivative of a scalar vector (tangent vector)

If x is the scalar, each element is

d(element)/dx

37

Orthogonal projection of a vector matrix

A=1/||u||^2 [(ux)^2, (ux)(uy)]
[(ux)(uy), (uy)^2 ]

38

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

39

Hessian Matrix

A square matrix of second order partial derivatives of a function

40

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

41

Eigenfunction

A transformation of a matrix for which a given vector in that plane, by definition the eigenvector, remains completely unchanged

42

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

43

Normal mode of a matrix transformation

The pattern (frequency and rmagnetude).by which a set of transformations repeat themselves within a given system

44

Minor of a Matrix element

The determinant of the matrix that excludes the elements in the rows or columns of that element