Matrix Calculus

Question 1

Determining a minor in a matrix

Answer 1

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

Question 2

Matrix determinant

Answer 2

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

Question 3

Starting position in a matrix

i=1, j=1

Answer 3

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

Question 4

Dimensions of a matrix, and propper notation for element identity

Answer 4

Rows-by-columns

Question 5

Referring to an individual item in a matrix

Answer 5

(Matrix-symbol-title)ij

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

Question 6

Matrix addition/subtraction

Answer 6

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

Question 7

Scalar multiplication of matrices

Answer 7

Multiply each element within the matrix by the scalar-constant

Question 8

Transposing matrices

Answer 8

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

Writen as [matrix]^T

Question 9

Size of a product matrix in Matrix Multiplication

Answer 9

Rows1 by columns2

Question 10

Matrix multiplication

Answer 10

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

Question 11

Row addition

Answer 11

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

Question 12

Row multiplication

Answer 12

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

Question 13

Row switching

Answer 13

Switching the position of two rows in a matrix

Question 14

Sub-matrices

Answer 14

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

Question 15

Linear equations from matrices

Answer 15

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)

Question 16

Square matrix

Answer 16

A matrix with the same number of rows as columns

Question 17

Diagonal matrices

Answer 17

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

Question 18

Triangular matrix

Answer 18

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

Question 19

Identity matrix

Answer 19

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

Question 20

Main diagonal within a matrix

Answer 20

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

Question 21

Symmetric Matrices

Answer 21

A matrix which is equal to its own transpose

Question 22

Skew-symmetric matrix

Answer 22

A matrix which is equal to its own NEGATIVE transpose

meaning the scalar-product of the transpose and -1

Question 23

Geometric shapes from 2x2 matrices

Answer 23

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

Question 24

Horizontal shear transformation for geometric figures by a 2x2 matrix

Answer 24

Element 12 increases

Question 25

Horizontal flip transformation for geometric figures by a 2x2 matrix

Answer 25

Element 11 is made negative

Question 26

Verticle flip transformation for geometric figures by a 2x2 matrix

Answer 26

Element 22 is made negative

Question 27

Squeeze flip transformation for geometric figures by a 2x2 matrix

Answer 27

Element 11 becomes the reciprocal of the fractional element 22

Question 28

Scaling transformation for geometric figures by a 2x2 matrix

Answer 28

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

Question 29

Rotational ransformation for geometric figures by a 2x2 matrix

Answer 29

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

Question 30

Inverse matrix

Answer 30

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

Question 31

Matrix functions

Answer 31

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

Question 32

Orthoganal Matrix

Answer 32

A matrix for which the transpose is equal to the inverse

Question 33

Matrix trace

Answer 33

Sum of its diagonal elements

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

Question 34

Gradient Matrix

Answer 34

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

Question 35

Matrices of a vector

Answer 35

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

Question 36

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

Answer 36

If x is the scalar, each element is

d(element)/dx

Question 37

Orthogonal projection of a vector matrix

Answer 37

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

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

Question 38

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

Answer 38

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

Write it out

Question 39

Hessian Matrix

Answer 39

A square matrix of second order partial derivatives of a function

Question 40

Eigenvector of matrix transformation

Answer 40

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

Question 41

Eigenfunction

Answer 41

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

Question 42

Use for transformations of a matrix

Answer 42

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

Question 43

Normal mode of a matrix transformation

Answer 43

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

Question 44

Minor of a Matrix element

Answer 44

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