SLIDE 1 Math 221: LINEAR ALGEBRA
Chapter 2. Matrix Algebra §2-1. Matrix Addition, Scalar Multiplication and Transposition
Le Chen1
Emory University, 2020 Fall
(last updated on 10/21/2020) Creative Commons License (CC BY-NC-SA) 1Slides are adapted from those by Karen Seyffarth from University of Calgary.
SLIDE 2 Matrices - Basic Definitions and Notation
Definition
Let m and n be positive integers. ◮ An m × n matrix is a rectangular array of numbers having m rows and n
- columns. Such a matrix is said to have size m × n.
General notation for an matrix, : . . . . . . . . . . . .
SLIDE 3 Matrices - Basic Definitions and Notation
Definition
Let m and n be positive integers. ◮ An m × n matrix is a rectangular array of numbers having m rows and n
- columns. Such a matrix is said to have size m × n.
◮ A row matrix (or row) is a 1 × n matrix, and a column matrix (or column) is an m × 1 matrix. General notation for an matrix, : . . . . . . . . . . . .
SLIDE 4 Matrices - Basic Definitions and Notation
Definition
Let m and n be positive integers. ◮ An m × n matrix is a rectangular array of numbers having m rows and n
- columns. Such a matrix is said to have size m × n.
◮ A row matrix (or row) is a 1 × n matrix, and a column matrix (or column) is an m × 1 matrix. ◮ A square matrix is an n × n matrix. General notation for an matrix, : . . . . . . . . . . . .
SLIDE 5 Matrices - Basic Definitions and Notation
Definition
Let m and n be positive integers. ◮ An m × n matrix is a rectangular array of numbers having m rows and n
- columns. Such a matrix is said to have size m × n.
◮ A row matrix (or row) is a 1 × n matrix, and a column matrix (or column) is an m × 1 matrix. ◮ A square matrix is an n × n matrix. ◮ The (i, j)-entry of a matrix is the entry in row i and column j. For a matrix A, the (i, j)-entry of A is often written as aij. General notation for an matrix, : . . . . . . . . . . . .
SLIDE 6 Matrices - Basic Definitions and Notation
Definition
Let m and n be positive integers. ◮ An m × n matrix is a rectangular array of numbers having m rows and n
- columns. Such a matrix is said to have size m × n.
◮ A row matrix (or row) is a 1 × n matrix, and a column matrix (or column) is an m × 1 matrix. ◮ A square matrix is an n × n matrix. ◮ The (i, j)-entry of a matrix is the entry in row i and column j. For a matrix A, the (i, j)-entry of A is often written as aij. General notation for an m × n matrix, A: A = a11 a12 a13 . . . a1n a21 a22 a23 . . . a2n a31 a32 a33 . . . a3n . . . . . . . . . . . . am1 am2 am3 . . . amn =
SLIDE 7
Matrices – Properties and Operations
two matrices are equal if and only if they have the same size and the corresponding entries are equal. an matrix with all entries equal to zero. matrices must have the same size; add corresponding entries. multiply each entry of the matrix by the scalar. for an matrix , its negative is denoted and . for matrices and , .
SLIDE 8 Matrices – Properties and Operations
- 1. Equality: two matrices are equal if and only if they have the same size
and the corresponding entries are equal. an matrix with all entries equal to zero. matrices must have the same size; add corresponding entries. multiply each entry of the matrix by the scalar. for an matrix , its negative is denoted and . for matrices and , .
SLIDE 9 Matrices – Properties and Operations
- 1. Equality: two matrices are equal if and only if they have the same size
and the corresponding entries are equal.
- 2. Zero Matrix: an m × n matrix with all entries equal to zero.
matrices must have the same size; add corresponding entries. multiply each entry of the matrix by the scalar. for an matrix , its negative is denoted and . for matrices and , .
SLIDE 10 Matrices – Properties and Operations
- 1. Equality: two matrices are equal if and only if they have the same size
and the corresponding entries are equal.
- 2. Zero Matrix: an m × n matrix with all entries equal to zero.
- 3. Addition: matrices must have the same size; add corresponding entries.
multiply each entry of the matrix by the scalar. for an matrix , its negative is denoted and . for matrices and , .
SLIDE 11 Matrices – Properties and Operations
- 1. Equality: two matrices are equal if and only if they have the same size
and the corresponding entries are equal.
- 2. Zero Matrix: an m × n matrix with all entries equal to zero.
- 3. Addition: matrices must have the same size; add corresponding entries.
- 4. Scalar Multiplication: multiply each entry of the matrix by the scalar.
for an matrix , its negative is denoted and . for matrices and , .
SLIDE 12 Matrices – Properties and Operations
- 1. Equality: two matrices are equal if and only if they have the same size
and the corresponding entries are equal.
- 2. Zero Matrix: an m × n matrix with all entries equal to zero.
- 3. Addition: matrices must have the same size; add corresponding entries.
- 4. Scalar Multiplication: multiply each entry of the matrix by the scalar.
- 5. Negative of a Matrix: for an m × n matrix A, its negative is denoted −A
and −A = (−1)A. for matrices and , .
SLIDE 13 Matrices – Properties and Operations
- 1. Equality: two matrices are equal if and only if they have the same size
and the corresponding entries are equal.
- 2. Zero Matrix: an m × n matrix with all entries equal to zero.
- 3. Addition: matrices must have the same size; add corresponding entries.
- 4. Scalar Multiplication: multiply each entry of the matrix by the scalar.
- 5. Negative of a Matrix: for an m × n matrix A, its negative is denoted −A
and −A = (−1)A.
- 6. Subtraction: for m × n matrices A and B, A − B = A + (−1)B.
SLIDE 14
Matrix Addition
Definition
Let A = [aij] and B = [bij] be two m × n matrices. Then A + B = C where C is the m × n matrix C = [cij] defined by cij = aij + bij
SLIDE 15 Matrix Addition
Definition
Let A = [aij] and B = [bij] be two m × n matrices. Then A + B = C where C is the m × n matrix C = [cij] defined by cij = aij + bij
Example
Let A = 1 3 2 5
−2 6 1
A + B = 1 + 0 3 + −2 2 + 6 5 + 1
1 1 8 6
SLIDE 16
Theorem (Properties of Matrix Addition)
Let A, B and C be m × n matrices. Then the following properties hold.
SLIDE 17 Theorem (Properties of Matrix Addition)
Let A, B and C be m × n matrices. Then the following properties hold.
- 1. A + B = B + A (matrix addition is commutative).
SLIDE 18 Theorem (Properties of Matrix Addition)
Let A, B and C be m × n matrices. Then the following properties hold.
- 1. A + B = B + A (matrix addition is commutative).
- 2. (A + B) + C = A + (B + C) (matrix addition is associative).
SLIDE 19 Theorem (Properties of Matrix Addition)
Let A, B and C be m × n matrices. Then the following properties hold.
- 1. A + B = B + A (matrix addition is commutative).
- 2. (A + B) + C = A + (B + C) (matrix addition is associative).
- 3. There exists an m × n zero matrix, 0, such that A + 0 = A.
(existence of an additive identity).
SLIDE 20 Theorem (Properties of Matrix Addition)
Let A, B and C be m × n matrices. Then the following properties hold.
- 1. A + B = B + A (matrix addition is commutative).
- 2. (A + B) + C = A + (B + C) (matrix addition is associative).
- 3. There exists an m × n zero matrix, 0, such that A + 0 = A.
(existence of an additive identity).
- 4. There exists an m × n matrix −A such that A + (−A) = 0.
(existence of an additive inverse).
SLIDE 21
Scalar Multiplication
Definition
Let A = [aij] be an m × n matrix and let k be a scalar. Then kA = [kaij].
SLIDE 22
Scalar Multiplication
Definition
Let A = [aij] be an m × n matrix and let k be a scalar. Then kA = [kaij].
Example
Let A = 2 −1 3 1 −2 4 5 .
SLIDE 23
Scalar Multiplication
Definition
Let A = [aij] be an m × n matrix and let k be a scalar. Then kA = [kaij].
Example
Let A = 2 −1 3 1 −2 4 5 . Then 3A = 3(2) 3(0) 3(−1) 3(3) 3(1) 3(−2) 3(0) 3(4) 3(5) = 6 −3 9 3 −6 12 15
SLIDE 24
Theorem (Properties of Scalar Multiplication)
Let A, B be m × n matrices and let k, p ∈ R (scalars). Then the following properties hold.
SLIDE 25 Theorem (Properties of Scalar Multiplication)
Let A, B be m × n matrices and let k, p ∈ R (scalars). Then the following properties hold.
(scalar multiplication distributes over matrix addition).
SLIDE 26 Theorem (Properties of Scalar Multiplication)
Let A, B be m × n matrices and let k, p ∈ R (scalars). Then the following properties hold.
(scalar multiplication distributes over matrix addition).
(addition distributes over scalar multiplication).
SLIDE 27 Theorem (Properties of Scalar Multiplication)
Let A, B be m × n matrices and let k, p ∈ R (scalars). Then the following properties hold.
(scalar multiplication distributes over matrix addition).
(addition distributes over scalar multiplication).
- 3. k (pA) = (kp) A. (scalar multiplication is associative).
SLIDE 28 Theorem (Properties of Scalar Multiplication)
Let A, B be m × n matrices and let k, p ∈ R (scalars). Then the following properties hold.
(scalar multiplication distributes over matrix addition).
(addition distributes over scalar multiplication).
- 3. k (pA) = (kp) A. (scalar multiplication is associative).
- 4. 1A = A. (existence of a multiplicative identity).
SLIDE 29 Example
2 −1 1 1
−2 1 3
6 8 1 −1
SLIDE 30 Example
2 −1 1 1
−2 1 3
6 8 1 −1
−16 −4 13 3
SLIDE 31 Example
2 −1 1 1
−2 1 3
6 8 1 −1
−16 −4 13 3
Let A and B be m × n matrices. Simplify the expression 2[9(A − B) + 7(2B − A)] − 2[3(2B + A) − 2(A + 3B) − 5(A + B)]
SLIDE 32 Example
2 −1 1 1
−2 1 3
6 8 1 −1
−16 −4 13 3
Let A and B be m × n matrices. Simplify the expression 2[9(A − B) + 7(2B − A)] − 2[3(2B + A) − 2(A + 3B) − 5(A + B)]
Solution
2[9(A − B) + 7(2B − A)] − 2[3(2B + A) − 2(A + 3B) − 5(A + B)] = 2(9A − 9B + 14B − 7A) − 2(6B + 3A − 2A − 6B − 5A − 5B) = 2(2A + 5B) − 2(−4A − 5B) = 12A + 20B
SLIDE 33
Definition (Matrix Transpose)
If A is an m × n matrix, then its transpose, denoted AT, is the n × m whose ith row is the ith column of A, 1 ≤ i ≤ n; i.e., if A = [aij], then AT = [aij]T = [aji] i.e., the (i, j)-entry of AT is the (j, i)-entry of A.
SLIDE 34
Definition (Matrix Transpose)
If A is an m × n matrix, then its transpose, denoted AT, is the n × m whose ith row is the ith column of A, 1 ≤ i ≤ n; i.e., if A = [aij], then AT = [aij]T = [aji] i.e., the (i, j)-entry of AT is the (j, i)-entry of A.
Theorem (Properties of the Transpose of a Matrix)
Let A and B be m × n matrices, C be a n × p matrix, and r ∈ R a scalar. Then
SLIDE 35 Definition (Matrix Transpose)
If A is an m × n matrix, then its transpose, denoted AT, is the n × m whose ith row is the ith column of A, 1 ≤ i ≤ n; i.e., if A = [aij], then AT = [aij]T = [aji] i.e., the (i, j)-entry of AT is the (j, i)-entry of A.
Theorem (Properties of the Transpose of a Matrix)
Let A and B be m × n matrices, C be a n × p matrix, and r ∈ R a scalar. Then
SLIDE 36 Definition (Matrix Transpose)
If A is an m × n matrix, then its transpose, denoted AT, is the n × m whose ith row is the ith column of A, 1 ≤ i ≤ n; i.e., if A = [aij], then AT = [aij]T = [aji] i.e., the (i, j)-entry of AT is the (j, i)-entry of A.
Theorem (Properties of the Transpose of a Matrix)
Let A and B be m × n matrices, C be a n × p matrix, and r ∈ R a scalar. Then
- 1. (AT)T = A
- 2. (rA)T = rAT
SLIDE 37 Definition (Matrix Transpose)
If A is an m × n matrix, then its transpose, denoted AT, is the n × m whose ith row is the ith column of A, 1 ≤ i ≤ n; i.e., if A = [aij], then AT = [aij]T = [aji] i.e., the (i, j)-entry of AT is the (j, i)-entry of A.
Theorem (Properties of the Transpose of a Matrix)
Let A and B be m × n matrices, C be a n × p matrix, and r ∈ R a scalar. Then
- 1. (AT)T = A
- 2. (rA)T = rAT
- 3. (A + B)T = AT + BT
SLIDE 38 Definition (Matrix Transpose)
If A is an m × n matrix, then its transpose, denoted AT, is the n × m whose ith row is the ith column of A, 1 ≤ i ≤ n; i.e., if A = [aij], then AT = [aij]T = [aji] i.e., the (i, j)-entry of AT is the (j, i)-entry of A.
Theorem (Properties of the Transpose of a Matrix)
Let A and B be m × n matrices, C be a n × p matrix, and r ∈ R a scalar. Then
- 1. (AT)T = A
- 2. (rA)T = rAT
- 3. (A + B)T = AT + BT
- 4. (AC)T = CTAT
SLIDE 39 Definition (Matrix Transpose)
If A is an m × n matrix, then its transpose, denoted AT, is the n × m whose ith row is the ith column of A, 1 ≤ i ≤ n; i.e., if A = [aij], then AT = [aij]T = [aji] i.e., the (i, j)-entry of AT is the (j, i)-entry of A.
Theorem (Properties of the Transpose of a Matrix)
Let A and B be m × n matrices, C be a n × p matrix, and r ∈ R a scalar. Then
- 1. (AT)T = A
- 2. (rA)T = rAT
- 3. (A + B)T = AT + BT
- 4. (AC)T = CTAT
To prove each these properties, you only need to compute the (i, j)-entries
- f the matrices on the left-hand side and the right-hand side.
SLIDE 40 Definition (Matrix Transpose)
If A is an m × n matrix, then its transpose, denoted AT, is the n × m whose ith row is the ith column of A, 1 ≤ i ≤ n; i.e., if A = [aij], then AT = [aij]T = [aji] i.e., the (i, j)-entry of AT is the (j, i)-entry of A.
Theorem (Properties of the Transpose of a Matrix)
Let A and B be m × n matrices, C be a n × p matrix, and r ∈ R a scalar. Then
- 1. (AT)T = A
- 2. (rA)T = rAT
- 3. (A + B)T = AT + BT
- 4. (AC)T = CTAT
To prove each these properties, you only need to compute the (i, j)-entries
- f the matrices on the left-hand side and the right-hand side. And you can
do it!
SLIDE 41 Problem
Find the matrix A if
1 −1 1 2 4 T = 2 1 5 3 8 .
SLIDE 42 Problem
Find the matrix A if
1 −1 1 2 4 T = 2 1 5 3 8 .
Solution
1 −1 1 2 4 TT = 2 1 5 3 8
T
SLIDE 43 Problem
Find the matrix A if
1 −1 1 2 4 T = 2 1 5 3 8 .
Solution
1 −1 1 2 4 TT = 2 1 5 3 8
T
A + 3 1 −1 1 2 4
2 3 1 5 8
SLIDE 44 Problem
Find the matrix A if
1 −1 1 2 4 T = 2 1 5 3 8 .
Solution
1 −1 1 2 4 TT = 2 1 5 3 8
T
A + 3 1 −1 1 2 4
2 3 1 5 8
= 2 3 1 5 8
1 −1 1 2 4
SLIDE 45 Problem
Find the matrix A if
1 −1 1 2 4 T = 2 1 5 3 8 .
Solution
1 −1 1 2 4 TT = 2 1 5 3 8
T
A + 3 1 −1 1 2 4
2 3 1 5 8
= 2 3 1 5 8
1 −1 1 2 4
= −1 3 3 −2 −1 −4
SLIDE 46
Symmetric Matrices
Definition
Let A = [aij] be an m × n matrix. The entries a11, a22, a33, . . . are called the main diagonal of A.
SLIDE 47
Symmetric Matrices
Definition
Let A = [aij] be an m × n matrix. The entries a11, a22, a33, . . . are called the main diagonal of A.
Definition
The matrix A is called symmetric if and only if AT = A. Note that this immediately implies that A is a square matrix.
SLIDE 48 Symmetric Matrices
Definition
Let A = [aij] be an m × n matrix. The entries a11, a22, a33, . . . are called the main diagonal of A.
Definition
The matrix A is called symmetric if and only if AT = A. Note that this immediately implies that A is a square matrix.
Examples
−3 −3 17
−1 5 2 11 5 11 −3 , 2 5 −1 2 1 −3 5 −3 2 −7 −1 −7 4 are symmetric matrices, and each is symmetric about its main diagonal.
SLIDE 49
Problem
Show that if A and B are symmetric matrices, then AT + 2B is symmetric.
SLIDE 50
Problem
Show that if A and B are symmetric matrices, then AT + 2B is symmetric.
Proof.
(AT + 2B)T = (AT)T + (2B)T
SLIDE 51
Problem
Show that if A and B are symmetric matrices, then AT + 2B is symmetric.
Proof.
(AT + 2B)T = (AT)T + (2B)T = A + 2BT
SLIDE 52
Problem
Show that if A and B are symmetric matrices, then AT + 2B is symmetric.
Proof.
(AT + 2B)T = (AT)T + (2B)T = A + 2BT = AT + 2B, since AT = A and BT = B
SLIDE 53
Problem
Show that if A and B are symmetric matrices, then AT + 2B is symmetric.
Proof.
(AT + 2B)T = (AT)T + (2B)T = A + 2BT = AT + 2B, since AT = A and BT = B Since (AT + 2B)T = AT + 2B, AT + 2B is symmetric.
SLIDE 54 Skew Symmetric Matrices
Definition
An n × n matrix A is said to be skew symmetric if AT = −A.
Example (Skew Symmetric Matrices)
−2
9 4 −9 −3 −4 3
SLIDE 55 Skew Symmetric Matrices
Definition
An n × n matrix A is said to be skew symmetric if AT = −A.
Example (Skew Symmetric Matrices)
−2
9 4 −9 −3 −4 3
Problem
Show that if A is a square matrix, then A − AT is skew-symmetric.
SLIDE 56 Skew Symmetric Matrices
Definition
An n × n matrix A is said to be skew symmetric if AT = −A.
Example (Skew Symmetric Matrices)
−2
9 4 −9 −3 −4 3
Problem
Show that if A is a square matrix, then A − AT is skew-symmetric.
Solution
We must show that (A − AT)T = −(A − AT).
SLIDE 57 Skew Symmetric Matrices
Definition
An n × n matrix A is said to be skew symmetric if AT = −A.
Example (Skew Symmetric Matrices)
−2
9 4 −9 −3 −4 3
Problem
Show that if A is a square matrix, then A − AT is skew-symmetric.
Solution
We must show that (A − AT)T = −(A − AT). Using the properties of matrix addition, scalar multiplication, and transposition (A − AT)T = AT − (AT)T = AT − A = −(A − AT).
