Chapter 1 Linear Equation and Matrices

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Chapter 1Linear Equation and Matrices
1.1 Systems of Linear Equations 1.2 Matrices
1.3 Matrix Multiplication 1.4 Algebraic
Properties of Matrix Operations 1.5 Special
Types of Matrices and Partitioned
Matrices 1.6 Matrix Transformations
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1.5 Special Matrices and Partitioned Matrices

• Defn - An n x n matrix A aij is called a
  diagonal matrix if aij 0 for i ? j, i.e. the
  terms off the main diagonal are all zero.
• Defn - A scalar matrix is a diagonal matrix whose
  diagonal elements are all equal.
• Defn - The scalar matrix In aij , where aii
  1 and aij 0 for i ? j is called the n x n
  identity matrix. The name comes from the
  following property. Let A be any m x n matrix,
  then A In A and Im A A.

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1.5 Special Matrices and Partitioned Matrices

• Matrix Powers
• Recall that matrix multiplication is associative,
  i.e. if A, B and C have the proper dimensions,
  then A ( BC ) ( AB ) C, so the parentheses
  are unnecessary and the product can be written as
  ABC.
• If A is an n x n matrix and p is a positive
  integer, can define
• Again, if A is an n x n matrix, adopt the
  convention

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1.5 Special Matrices and Partitioned Matrices

• Matrix Powers
• The following laws of exponents hold for
  nonnegative integers p and q and any n x n matrix
  A
• 1) Ap Aq Ap q
• 2) ( Ap ) q Apq
• Caution Without additional assumptions on A and
  B, we cannot do the following
• 1) define Ap for negative integers p
• 2) assert that ( AB ) p Ap B p

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1.5 Special Matrices and Partitioned Matrices

• Triangular Matrices
• An n x n matrix A aij is called upper
  triangular if aij 0 for i gt j
• An n x n matrix A aij is called lower
  triangular if aij 0 for i lt j
• Note
• A diagonal matrix is both upper and lower
  triangular
• The n x n zero matrix is both upper and lower
  triangular

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1.5 Special Matrices and Partitioned Matrices

• Symmetry
• Defn - A matrix A is called symmetric if AT A
• Defn - A matrix A is called skew-symmetric if AT
  -A
• Comment - If A is skew-symmetric, then the
  diagonal elements of A are zero
• Comment - Any square matrix A can be written as
  the sum of a symmetric matrix and a
  skew-symmetric matrix

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1.5 Special Matrices and Partitioned Matrices

• Partitioning of Matrices
• Defn - Let A aij be an m x n matrix. A
  submatrix of A is obtained by deleting some, but
  not all, of the rows and columns of A
• Example - Let
• some submatrices of A are

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1.5 Special Matrices and Partitioned Matrices

• Partitioning of Matrices
• Primary interest is in submatrices obtained by
  partitioning, i.e. by drawing horizontal and
  vertical lines between rows and columns of a
  matrix. Consider

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Special Matrices and Partitioned Matrices

• Partitioning of Matrices
• A can be written as where

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1.5 Special Matrices and Partitioned Matrices

• Partitioning of Matrices
• A could also be partitioned as

(Note Definitions of Aij have changed from
previous slide)
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1.5 Special Matrices and Partitioned Matrices

• Defn - An n x n matrix A is called nonsingular or
  invertible if there exists an n x n matrix B such
  that
• AB BA In .
• Comments
• If B exists, then B is called the inverse of A.
• If B does not exist, then A is called singular or
  noninvertible.
• At this point, the only available tool for
  showing that A is nonsingular is to show that B
  exists.

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1.5 Special Matrices and Partitioned Matrices

• Nonsingular Matrices
• Theorem - If the inverse of a matrix exists, then
  that inverse is unique.
• Proof - Let A be a nonsingular n x n matrix and
  let B and C be inverses of A. Then AB BA In
  and AC CA In B B In B( AC ) ( BA
  )C In C C
• so the inverse is unique.
• Notation - If A is a nonsingular matrix. The
  inverse of A is denoted by A-1.
• Comment - For nonsingular matrices, A, can define
  A raised to a negative power as A-k ( A-1 ) k
  k gt 0.

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1.5 Special Matrices and Partitioned Matrices

• Nonsingular Matrices
• Theorem - If A and B are both nonsingular
  matrices, then the product AB is nonsingular and
  ( AB ) -1 B-1A-1 .
• Proof - Consider the following products
• AB ( B-1A-1 ) AB B-1A-1 A In A-1 AA-1 In
• ( B-1A-1 ) AB B-1A-1AB B-1In B B-1B In
• Since we have found a matrix C such that
• C ( AB ) ( AB ) C In
• AB is nonsingular and its inverse is C B-1A-1 .

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1.5 Special Matrices and Partitioned Matrices

• Nonsingular Matrices
• Theorem - If A1, A2, , Ar are nonsingular
  matrices, then A1 A2 Ar is nonsingular and

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1.5 Special Matrices and Partitioned Matrices

• Nonsingular Matrices
• Theorem - If A is a nonsingular matrix, then A-1
  is nonsingular and ( A-1 ) -1 A .
• Proof - Since A-1 A A A-1 In , then A-1 is
  nonsingular and its inverse is A. So ( A-1 ) -1
  A .

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1.5 Special Matrices and Partitioned Matrices

• Nonsingular Matrices
• Comments
• Have observed earlier that AB AC does not
  necessarily imply that B C. However, if A is an
  n x n nonsingular matrix and AB AC, then B C.
  
• AB AC ? A-1 AB A-1 AC ? B C .
• Have observed earlier that AB 0 does not imply
  that A 0 or B 0. However, if A is an n x n
  nonsingular matrix and AB 0, then B 0 .
• A-1 ( AB ) A-1 0 ? ( A-1A ) B 0 ? B 0

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1.5 Special Matrices and Partitioned Matrices

• Nonsingular Matrices
• Theorem - If A is a nonsingular matrix, then AT
  is nonsingular and ( AT ) -1 ( A-1 ) T
• Proof - By an earlier theorem, ( AB )T BTAT
  for any two matrices A and B. Since A is
  nonsingular,
• A-1 A A A-1 In . Applying the relationship
  on transposes gives
• AT ( A-1 )T ( A-1 A )T InT In
• ( A-1 )T AT ( AA-1 )T InT In
• Since AT ( A-1 )T In and ( A-1 )T AT In , AT
  is nonsingular and its inverse is ( A-1 )T , i.e.
  
• ( AT ) -1 (A-1) T

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1.5 Special Matrices and Partitioned Matrices

• Linear Systems and Inverses
• A system of n linear equations in n unknowns may
  be written as Ax b, where A is n x n matrix. If
  A is nonsingular, then A-1 exists and the system
  may be solved by multiplying both sides by A-1
• A-1( Ax ) A-1b ? ( A-1A )x A-1b ? x A-1b

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1.5 Special Matrices and Partitioned Matrices

• Linear Systems and Inverses
• Comment - Although x A-1b gives a simple
  expression for the solution, its primary usage is
  for proofs and derivations.
• At this point we have no practical tool for
  computing A-1.
• Even with a tool for computing A-1, this method
  of solution is usually numerically inefficient.
  The only exception is if A has a special
  structure that lets A-1 have a simple
  relationship to A.