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EE 230 Lecture 1

Ran Cheng

Department of Electrical and Computer Engineering

Department of Physics and Astronomy University of California, Riverside

Overview

Textbook: Linear Algebra and Its Applications, 4th Ed, Gilbert Strang

(Cover Chapter 1 to 6 and part of Chapter 7.)

Contents: Advanced concepts and techniques in Linear Algebra

Classes focus on the fundamental aspects;

Discussions provide coding, more exercises, and homework Q&A

Lecture notes will be uploaded AFTER class

Matrix multiplication

[AB]ij =

k

aik bkj ≡ aik bki

Einstein convention: repeated indices are automatically summed

Number of rows (columns) of A = Number of columns (rows) of B

Matrix multiplication are not always commutative!

Example

Pauli matrices: σx σz 6 = σz σx (In fact, σx σz = −σz σx )

If AB = BA, or [A, B] = 0, commutative or Abelian (special cases)

If AB = −BA, or {A, B} = 0, anti-commutative

In general, AB 6 = ±BA: Matrix multiplication is non-Abelian!

Association: (AB)C = A(BC )

Distribution: A(B + C ) = AB + AC and (B + C )D = BD + CD

Transpose: (AB)

T = B

T A

T ; complex (AB)

H = B

H A

H

If multiple matrices, (ABC · · · )

T = · · · C

T B

T A

T

More than two indices? e.g., aijkmbkmlp = cijlp. Tensor Analysis

Trace of square matrix (m = n)

TrA =

i

aii summing all diagonal elements

Tr(A + B) = TrA + TrB

Tr(cA) = cTrA (c is a number)

TrA = TrA

T (not A

H !)

Tr(AB) = Tr(BA)

Tr(ABC ) 6 = Tr(ACB), but Tr(ABC ) = Tr(CAB)

Tr(ABC · · · X Y Z ) = Tr(Z ABC · · · X Y ) = Tr(Y Z ABC · · · X )

Cyclic property: permutation operations preserve Trace!

det A = det A

T

det(AB) = det A det B (Homework)

det(A + B) 6 = det A + det B

Changes sign under row or column exchange

∣ ∣ ∣ ∣

a b

c d

c d

a b

d c

b a

Vanishes if any two rows (columns) are proportional (as |A| = −|A|)

Decomposition with respect to only one row (column)

∣ ∣ ∣ ∣

a + a

′ b + b

′

c d

a b

c d

a

′ b

′

c d

Subtraction of a multiple of another row (column)

∣ ∣ ∣ ∣

a − kc b − kd

c d

a b

c d

Applications in Physics: Volume

Inverse matrix

A

− 1 A = AA

− 1 = I (square matrix)

Invertible for det A 6 = 0

(A

− 1 )

− 1 = A

(kA)

− 1

1 k

A

− 1 for k 6 = 0

(A

T )

− 1 = (A

− 1 )

T

If A = A

T and A

− 1 exists, then (A

− 1 )

T = A

− 1

(ABC · · · )

− 1 = · · · C

− 1 B

− 1 A

− 1

det(A

− 1 ) =

1 det A

Computation of inverse matrix

A

− 1

adjA

|A|

Find the determinant |A|

Minor Mij : the determinant of the (n − 1) × (n − 1) submatrix by

deleting row-i and column-j

Cofactor cij = (−1)

i+j Mij with 0 ≤ i, j ≤ n

Adjugate adj(A) = C

T by transposing the matrix of cofactors

Divide by the determinant: A

− 1 = adjA/|A|

(They are standard procedures)

Cofactor expansion

det A =

n ∑

j=

aij Cij =

n ∑

j=

i+j aij det Mij , for ∀ i (1)

Similar for column expansion, det A =

∑n

i=1 aij^ Cij^ for^ ∀^ j

Because AC

T = (det A)I , we know

For wrong combinations

Amounts to expanding det of a matrix with two identical rows!

Example: Find the determinant of the n × n matrix An

Solution: Use cofactor expansion |An| = 2C 11 + (−1)C 12

C 11 = |An− 1 |, C 12 = (−1)

1+

= +|An− 2 | + 0

Recursion relation: |An| = 2|An− 1 | − |An− 2 |

Since |A 1 | = 2, |A 2 | = 3 · · · , we can solve |An| = n + 1

Example: Use Cramer’s rule to solve

Solution: The coefficient matrix in non-singular, |A| = 13.

References

Linear Algebra and Its Applications, 4th Ed., Gilbert Strang

Applied Linear Algebra, 2nd Ed., Peter J. Olver and Chehrzad Shakiban

Matrix Analysis and Applied Linear Algebra, Carl D. Meyer

Advanced Engineering Mathematics, 7th Ed., Peter V. O’Neil