compiled by raj g. tiwari. vector: n × 1 matrix interpretation: a point or line in n- dimensional...
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Compiled By
Raj G. Tiwari
Vector: n×1 matrix Interpretation:
a point or line in n-dimensional space
Dot Product, Cross Product, and Magnitude defined on vectors only
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b
a
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cfbead
f
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cbaABBA T
ccbbaaAAA T 2
Think of the dot product as a matrix multiplication
The magnitude is the dot product of a vector with itself
The cross-product can be computed as a specially constructed determinant
A
B
A×B
xyyx
zxxz
yzzy
zyx
zyx
baba
baba
baba
bbb
aaa
kji
BA
ˆˆˆ
A matrix is a set of elements, organized into rows and columns
dc
barows
columns
Transpose: Swap rows with columns
ihg
fed
cba
M
ifc
heb
gda
M T
z
y
x
V zyxV T
Addition, Subtraction, Multiplication
hdgc
fbea
hg
fe
dc
ba
hdgc
fbea
hg
fe
dc
ba
dhcfdgce
bhafbgae
hg
fe
dc
ba
Just add elements
Just subtract elements
Multiply each row by each
column
Is AB = BA? Maybe, but maybe not!
Heads up: multiplication is NOT commutative!
Exceptions AB=BA iff
B = a scalar,B = identity matrix I, orB = the inverse of A, i.e., A-1
......
...bgae
hg
fe
dc
ba
......
...fcea
dc
ba
hg
fe
Multiplication of matrices require conformability condition
The conformability condition for multiplication is that the column dimensions of the lead matrix A must be equal to the row dimension of the lag matrix B.
An m×n can be multiplied by an n×p matrix to yield an m×p result
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A symmetric matrix is a square matrix that is equal to its transpose
A=At
The entries of a symmetric matrix are symmetric with respect to the main diagonal (top left to bottom right). So if the entries are written as A = (aij), then
aij=aji
For Example
A skew-symmetric (or antisymmetric or antimetric) matrix is a square matrix A whose transpose is also its negative; that is, it satisfies the equation A = −AT. If the entry in the i th row and j th column is aij, i.e. A = (aij) then the symmetric condition becomes aij = −aji. For example, the following matrix is skew-symmetric:
000
000
000
.
100
010
001
10
01
etc
or Identity Matrix is a square matrix and also it is a diagonal matrix with 1 along the diagonals similar to scalar “1”
Null matrix is one in which all elements are zero
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6.837 Linear Algebra Review
Used for inversion If det(A) = 0, then A
has no inverse Can be found using
factorials, pivots, and cofactors!
dc
baA
bcadAA )det(
If M is our d × d matrix, we define Mi|j to be the (d − 1) × (d − 1) matrix obtained by deleting the ith row and the jth column of M:
cegbdiafhcdhbfgaei
cegcdhbfgbdiafhaei
hg
edc
ig
fdb
ih
feaA
)(
)det(
ihg
fed
cba
ihg
fed
cba
ihg
fed
cbaFor a 3×3 matrix:Sum from left to rightSubtract from right to left
Note: In the general case, the determinant has n! terms
ihg
fed
cbaFor Matrix A
252
314
231
Let's expand our matrix along the first row. From the sign chart, we see that 1 is in a positive position, 3
is in a negative position and 2 is in a positive position. By putting the + or - in front of the element, it takes care of the sign adjustment when going from the minor to the cofactor.
1 ( 2 - 15 ) - 3 ( 8 - 6 ) + 2 ( 20 - 2 )= 1 ( -13 ) - 3 ( 2 ) + 2 (18)= -13 - 6 + 36= 17
Det(A)=
The term Mij is known as the ”minor matrix” and is the matrix you get if you eliminate row i and column j from matrix A.
Minor matrix calculation Minor matrix
Cofactor matrix
Adjoint matrix
Adjoint can be found by transposing the matrix of cofactors
6.837 Linear Algebra Review
Identity matrix: AI = A
Some matrices have an inverse, such that:AA-1 = I
Inversion is tricky:(ABC)-1 = C-1B-1A-1
Derived from non-commutativity property
100
010
001
I
Let A be a non-singular matrix. If there exists a square matrix B such that AB = I (identity matrix) then B is called inverse of matrix A and is denoted as A-1. i.e AA-1 = I Example:Matrix A Matrix B =Identity(I)1 3 1 x 2 9 -5 = 1 0 01 1 2 0 -2 1 0 1 02 3 4 -1 -3 2 0 0 1
It is not possible to divide one matrix by another. That is, we can not write A/B. This is because for two matrices A and B, the quotient can be written as AB-1
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The matrix must be square (same number of rows and columns).
The determinant of the matrix must not be zero (determinants are covered in section 6.4). This is instead of the real number not being zero to have an inverse, the determinant must not be zero to have an inverse.
A square matrix that has an inverse is called invertible or non-singular. A matrix that does not have an inverse is called singular.
A matrix does not have to have an inverse, but if it does, the inverse is unique.
The trace of a d × d (square) matrix, denoted tr[M], is the sum of its diagonal elements:
Let A be a square matrix. A non-zero vector X is called an eigenvector of A if and only if there exists a number (real or complex) such that
AX= λ X
If such a number exists, it is called an eigenvalue of A. The vector C is called eigenvector associated to the eigenvalue .
Remark. The eigenvector C must be non-zero since we have
for any number .
Rewriting (A- λI)X=0
In linear algebra, the characteristic equation (or secular equation) of a square matrix A is the equation in one variable λ
where det is the determinant and I is the identity matrix. The solutions of the characteristic equation are precisely the eigenvalues of the matrix A
Set
corresponding to an eigenvalue λ, we simply solve the system of linear equations given by
(A- λI)X=0
Applying characteristic equation
If we develop this determinant using the third column, we obtain
Using easy algebraic manipulations, we get which implies that the eigenvalues
of A are 0, -4, and 3.
Rewritten as
By Solving
where c is an arbitrary number
Let x and y be vectors of orders n and m respectively:
where each component yi may be a function of all the xj , a fact represented by saying that y is a function of x, or
y = y(x).
Derivative of a Scalar with Respect to Vector
Derivative of Vector with Respect to Scalar
If we have an m-dimensional vector-valued function of a n-dimensional vector x, we calculate the derivatives and represent them as the Jacobian Jacobian matrix
This matrix is also denoted by and .
The Jacobian determinant (often simply called the Jacobian) is the determinant of the Jacobian matrix.
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012
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