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B) Calculus > Matrix Calculus > Flashcards

Matrix Calculus Flashcards

0
Q

Determining a minor in a matrix

A

Remove the elements in the target’s row and column, including the target itself
Calculate the determinant of the new matrix

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1
Q

Minor of a Matrix element

A

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

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2
Q

Matrix determinant

A

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

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3
Q

Starting position in a matrix

i=1, j=1

A

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

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4
Q

Dimensions of a matrix, and propper notation for element identity

A

Rows-by-columns

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5
Q

Referring to an individual item in a matrix

A

(Matrix-symbol-title)ij

Where i is rows from top and j is columns from left

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6
Q

Matrix addition/subtraction

A

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

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7
Q

Scalar multiplication of matrices

A

Multiply each element within the matrix by the scalar-constant

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8
Q

Transposing matrices

A

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

Writen as [matrix]^T

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9
Q

Size of a product matrix in Matrix Multiplication

A

Rows1 by columns2

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10
Q

Matrix multiplication

A

Sum of the products of the next element in matrix1 row and matrix2 column corresponding to the element location in the product matrix

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11
Q

Row addition

A

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

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12
Q

Row multiplication

A

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

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13
Q

Row switching

A

Switching the position of two rows in a matrix

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14
Q

Sub-matrices

A

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

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15
Q

Linear equations from matrices

A

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)

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16
Q

Square matrix

A

A matrix with the same number of rows as columns

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17
Q

Diagonal matrices

A

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

18
Q

Triangular matrix

A

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

19
Q

Identity matrix

A

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

20
Q

Main diagonal within a matrix

A

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

21
Q

Symmetric Matrices

A

A matrix which is equal to its own transpose

22
Q

Skew-symmetric matrix

A

A matrix which is equal to its own NEGATIVE transpose

meaning the scalar-product of the transpose and -1

23
Q

Geometric shapes from 2x2 matrices

A

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
Q

Horizontal shear transformation for geometric figures by a 2x2 matrix

A

Element 12 increases

25
Q

Horizontal flip transformation for geometric figures by a 2x2 matrix

A

Element 11 is made negative

26
Q

Verticle flip transformation for geometric figures by a 2x2 matrix

A

Element 22 is made negative

27
Q

Squeeze flip transformation for geometric figures by a 2x2 matrix

A

Element 11 becomes the reciprocal of the fractional element 22

28
Q

Scaling transformation for geometric figures by a 2x2 matrix

A

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

29
Q

Rotational ransformation for geometric figures by a 2x2 matrix

A 
Elements 11 and 22 are multiplied by cos(angle)
While sin(angle) is added to element 21 and subtracted from element 12
30
Q

Inverse matrix

A

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
Q

Matrix functions

A

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
Q

Orthoganal Matrix

A

A matrix for which the transpose is equal to the inverse

33
Q

Matrix trace

A

Sum of its diagonal elements

Writen as tr(A), where ‘A’ is the matrix

34
Q

Gradient Matrix

A

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

35
Q

Matrices of a vector

A

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

36
Q

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

A

If x is the scalar, each element is

d(element)/dx

37
Q

Orthogonal projection of a vector matrix

A

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

[(ux)(uy), (uy)^2 ]

38
Q

Reflecting a vector/matrix about a line that goes through the origin

A

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

Write it out

39
Q

Hessian Matrix

A

A square matrix of second order partial derivatives of a function

40
Q

Eigenvector of matrix transformation

A

The vector within that plane the direction and position of which remains completely unchanged by the transformation, but about which the transformation occurs

41
Q

Eigenfunction

A

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

42
Q

Use for transformations of a matrix

A

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
Q

Normal mode of a matrix transformation

A

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

B) Calculus (17 decks)

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