CPSC 491
Xin Liu
Nov 17, 2010
Introduction
• Xin Liu
• PhD student of Dr. Rokne
• Contact
• liuxin@ucalgary.ca
• Slides downloadable at
• pages.cpsc.ucalgary.ca/~liuxin
• The way to math world
• Lecture attendance
• Hard to learn by yourselves
• Practices, practices, and practices …
2
Matrix-Vector
Multiplication
• Linear (the 1st degree) systems are the simplest, but most widely
used systems in science and engineering
• A basic problem: solving the linear equation system
• Straight forward method
• Gaussian elimination
• Hard to do because
• large scale
• poor conditioned
•
small disturbance in coefficients causes big difference in solutions
• A better method
• SVD – Singular Vale Decomposition
• Will be introduced gradually in a series of lectures
3
Definitions
• An n-vector is defined as
•
Think about 3-vectors in Euclidean space
• An mxn matrix is defined as
• Multiplication
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Linear Mapping
•
is a linear mapping, which satisfies
•
•
Distributive law
Associative law (for scalar)
• Conversely, every linear map from Rn to Rm can be expressed as a
multiplication by an mxn matrix
•
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Mat-vect multiplication
•
View the matrix-vector multiplication from another angle
•
If we write A as a combination of column vectors
•
Then the mat-vect multiplication can be written as
•
That means: b is a linear combination of the columns of A
•
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Mat-mat multiplication
•
Matrix-matrix multiplication
•
is defined as
•
We can calculate B columnwisely
•
Each column of B is a linear combination of the columns aj with the coefficients ckj
•
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Range
• Definition:
• The range of a matrix A, is the set of vectors that
can be expressed as Ax for some x.
• Theorem
• range (A) is the space spanned by the columns of A.
• The range of A is also called the column space of A.
8
Nullspace
• Definition:
• The nullspace (solution space) of A is the set of
vectors x that satisfy Ax = 0.
• Each vector x in the nullspace gives the
expansion coefficients of the zero vector as a
linear combination of columns of A
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Rank
•
Column rank = dimension of space spanned by the matrix’s columns = # of linearly
independent columns
•
Row rank = dimension of space spanned by the matrix’s rows = # of linearly
independent rows
•
Row rank = Column rank = Matrix rank
•
Full rank
•
Theorem
•
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Inverse
•
A nonsingular or invertible matrix must be square and full rank.
•
The m columns of a nonsingular mxm matrix A span (form a basis) for the whole space
Rm
•
•
Any vector in Rm can be expressed as a linear combination of the columns of A
The inverse of A is a matrix A-1, such that
•
•
AA-1 = A-1A = I
I is the mxm identity matrix
•
The inverse of a nonsingular matrix is unique.
•
A-1b is the unique solution of Ax = b.
•
A-1b is the vector of coefficients of the expansion of b in the basis of the columns of A.
•
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Transpose
• Definition
• The transpose AT of an mxn matrix A is nxm where the (i,j) entry of AT
is the (j, i) entry of A.
• Example
• A is symmetric if A = AT.
• Multiplication
•
•
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Inner product
• Inner product
• Euclidean length
• Angle
•
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Orthogonal vectors
• Orthogonal (perpendicular) vectors
•
Vectors x, y are orthogonal if xTy = 0.
• Orthogonal vector set
• Orthogonal two vector sets
•
14
Orthonormal
• Definition
• Theorem
• Corollary
•
15
Components of a vector
• Inner products can be used to decompose arbitrary vectors into
orthogonal components (project onto orthonormal vectors).
•
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Components of a vector
17
Orthogonal matrices
• Definition:
• According to the definition
• Or
•
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An example
• 2D rotation matrix
19
Multiplication by an
orthogonal matrix
• inner products is preserved
• angles between vectors are preserved
• lengths are preserved
•
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