ME751
Advanced Computational
Multibody Dynamics
Review: Elements of Linear Algebra & Calculus
September 9, 2016
Dan Negrut
University of Wisconsin-Madison
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ME751 Advanced Computational Multibody Dynamics, Lecture notes of Vector Analysis
University of Wisconsin-Madison ... Review of elements of Calculus (two definitions and three theorems) ... A bold lower case letter denotes a vector.
Typology: Lecture notes
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ME751Advanced ComputationalMultibody Dynamics Review: Elements of Linear Algebra & CalculusSeptember 9, 2016 Dan NegrutUniversity of Wisconsin-Madison
Quote of the day
If you can't convince them, confuse them.- Harry S. Truman (US President)
Notation Conventions^ ^ A bold upper case letter denotes matrices^ ^ Example:^ A
,^ B , etc. A bold lower case letter denotes a vector Example: v ,^ s , etc. A letter in italics format denotes a scalar quantity Example:ܽ ,ܾ
Matrix Review^ ^ Matrix: a tableau of elements^ ^ Matrix addition:^ ^ Addition is commutative
11 12 1 21 22 2
(^1221) 1 2
T T T n n n m m m^
m a^ a^ a a^ a^ a a^ a^ an
é^ ù é^ ù^
ê^ ú ê^ ú^
ê^ ú ê^ ú^
ê^ ú ê^ ú^ é
ù =^ =
= ê^
ú ¼ ¼^ ê
ú^ ê^ ¼ ¼ ¼ ¼ ¼ úë û^ ê^ ú ê^ ú^
ê^ ú ê^ ú^
ê^ ú ê^ úë^ û^
ë^ û A^
a aa a a^ a mn [ ] , 1 , (^1) a i m j^ ij mn [ ] , 1 , (^1) ij mn [ ] , ij ij iij j + = + A B B A n b^ i^ m^ j^ n c^ c^ a^ b ^ ^
^ ^ ^ ^
^ ^ ^ ^ ^ ^
^
A^ B C^ A^ B
Matrix-Vector Multiplication^ ^ A column-wise perspective on matrix-vector multiplication^ ^ Example:^ ^ A row-wise perspective on matrix-vector multiplication:
11 12 1 1
1 21 22 2 2
2 1 21 n 1 2
n n
n^ i^ i i m^ m^ mn^ n^
n a^ a^ a^ v^
v a^ a^ a^ v^
v^ v a^ a^ a^ v^
= v é^ ù é^ ù^
é^ ù ê^ ú ê^ ú^
ê^ ú ê^ ú ê^ ú^
ê^ ú ê^ ú ê^ ú^
ê^ úé ù =^
=^
ê^ ú ê^ ú^
ê^ úê úë û ê^ ú ê^ ú^
ê^ ú ê^ ú ê^ ú^
ê^ ú ê^ ú ê^ ú^
ê^ ú ë^ û ë^ û^
ë^ û ¼ ¼^
¼^ ¼^ ¼^ ¼ ¼
å
Av^
a^ a^ a^
a ^
^1
T^ T é ù^ é^ ù va^ a^1 ê ú^ ê^ ú T^ T ê ú^ ê^ ú va^ a ê ú^ ê^ ú^2 2 = = A v^ v ê ú^ ê^ ú ^ ê ú^ ê^ ú ê ú^ ê^ ú T^ T^ va^ a ê ú^ ê^ ú m^ m ë û^ ë^ û 1 4 2 0 1
(1)^ (2)^ ( 1)
·^ ·^ ·^0 1
^ ^ ^
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^ ^
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^
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^ ^
^
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Av
Matrix Review
[Cntd.]
^ Scaling of a matrix by a real number: scale each entry of the matrix^ ^ Example: ^ Transpose of a matrix
A^ dimension݉^ ݊ൈ^
T^ : a matrix B = A of dimension
݊ ݉ൈ^ whose^ ሻ ݆,݅ሺ^
entry is the^ ሻ ݅,݆ሺ^ entry of original matrix
A^8 ·^ ·[^ ]^ [^ ·^ ] a^ a ^ A ij^ ij 1 4 2 0 1.5^6
3 0 2 3 1 1 3
4.5^ 1.5^ 1. (1.5)^1 0 1 ^ ^ ^ ^ ^ ^ ^ ^ ^ ·1.5^0 1.5^ 1.5^ ^ ^ ^ ^ ^ ^ ^ ^ ^0 1 1 2 0 1.^5 1.5^3 ^ ^ ^ ^ ^ ^ ^ T 1 4 2 0 1 2 1 0 2 3 1 1 4 3 0 1 1 0 1 1 2 1 1 1 0 1 1 2 0 1 1 2 ^ ^ ^
^ ^ ^
^ ^ ^
^ ^ ^
^ ^
^ ^ ^
^ ^ ^ ^ ^
Matrix Rank^ ^ Row rank of a matrix^ ^ Largest number of rows of the matrix that are linearly independent^ ^ A matrix is said to have full row rank if the rank of the matrix is equal tothe number of rows of that matrix^ ^ Column rank of a matrix^ ^ Largest number of columns of the matrix that are linearly independent^ ^ Important results^ ^ For any matrix, the row rank and column rank are the same^ ^ This number is simply called the rank of the matrix^ ^ It follows that
Matrix Rank, Example ^ What is the row rank of the matrix
J? ^ What is the rank of
J?
2 1 1 0 4 2 2 1 ^ J 0 4 0 1
Matrix & Vector Norms, Example^ ^ Find norm 1, Euclidian, and Infinity for the following matrix:
1 2 A ^3 4
Matrix Review
[Cntd.] ^ Symmetric matrix: a square matrix
T A for which A = A
^ Skew-symmetric matrix: a square matrix
B^ for which^ B =- B
T
^ Examples: ^ Singular matrix: square matrix whose determinant is zero ^ Inverse of a square matrix
A : a matrix of the same dimension, called
-1 A ,
that satisfies the following:
^
^ ^
^
^ ^
^
^
^ ^
^
^ ^
^
^
^
^ ^
^
A^
B
Orthogonal & Orthonormal Matrices[we’ll work w/ a lot of orthonormal matrices]^ ^ Definition (
Q , orthogonal matrix): a square matrix
Q^ is
orthogonal if the product
T QQ^ is a diagonal matrix
^ Matrix^ Q^ is called orthonormal if it’s orthogonal and also
T QQ = I n
^ Note that people in general don’t make a distinction between an orthogonal andorthonormal matrix Note that if^ Q^ is an orthonormal matrix, then
-1^ T Q = Q
^ Example, orthonormal matrix:
Remark:On the Columns of an Orthonormal Matrix^ ^ Assume^ Q^ is an orthonormal matrix
^ In other words, the columns (and the rows) of an orthonormal matrixhave unit norm and are mutually perpendicular to each other