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

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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

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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

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A matrix which is equal to its own transpose

22

## Skew-symmetric matrix

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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

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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

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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

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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