Flashcards in Matrix Calculus Deck (44):

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## Minor of a Matrix element

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

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## Determining a minor in a matrix

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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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## 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

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##
Starting position in a matrix

(i=1, j=1)

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

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## Dimensions of a matrix, and propper notation for element identity

### Rows-by-columns

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## Referring to an individual item in a matrix

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(Matrix-symbol-title)ij

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

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## Matrix addition/subtraction

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

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## Scalar multiplication of matrices

### Multiply each element within the matrix by the scalar-constant

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## 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

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## Size of a product matrix in Matrix Multiplication

### Rows1 by columns2

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## 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

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## Row addition

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

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## Row multiplication

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

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## Row switching

### Switching the position of two rows in a matrix

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## Sub-matrices

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

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## 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)

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## Square matrix

### A matrix with the same number of rows as columns

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## Diagonal matrices

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

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## Triangular matrix

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

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## Identity matrix

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

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## Main diagonal within a matrix

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

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## Symmetric Matrices

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A matrix which is equal to its own transpose

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## 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

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## Geometric shapes from 2x2 matrices

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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

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## Horizontal shear transformation for geometric figures by a 2x2 matrix

### Element 12 increases

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## Horizontal flip transformation for geometric figures by a 2x2 matrix

### Element 11 is made negative

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## Verticle flip transformation for geometric figures by a 2x2 matrix

### Element 22 is made negative

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## Squeeze flip transformation for geometric figures by a 2x2 matrix

### Element 11 becomes the reciprocal of the fractional element 22

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## Scaling transformation for geometric figures by a 2x2 matrix

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

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## Rotational ransformation for geometric figures by a 2x2 matrix

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Elements 11 and 22 are multiplied by cos(angle)

While sin(angle) is added to element 21 and subtracted from element 12

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## Inverse matrix

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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

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## 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

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## Orthoganal Matrix

### A matrix for which the transpose is equal to the inverse

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## Matrix trace

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Sum of its diagonal elements

Writen as tr(A), where 'A' is the matrix

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## Gradient Matrix

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A matrix formed from the differentials that describe the vector gredient

Usually 1x3

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## Matrices of a vector

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

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

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If x is the scalar, each element is

d(element)/dx

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## Orthogonal projection of a vector matrix

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

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

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

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

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

Write it out

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## Hessian Matrix

### A square matrix of second order partial derivatives of a function

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## 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

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## Eigenfunction

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

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## Use for transformations of a matrix

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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

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