CPSC 491Xin Liu

Nov 17, 2010

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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 …

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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

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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.

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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

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Orthonormal

• Definition

• Theorem

• Corollary

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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

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Orthogonal matrices

• Definition:

• According to the definition

• Or

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An example

• 2D rotation matrix

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Multiplication by an orthogonal matrix

• inner products is preserved

• angles between vectors are preserved

• lengths are preserved

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