Formula Sheets: Linear Algebra | Engineering Mathematics for Electrical Engineering - Electrical Engineering (EE) PDF Download

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Engineering Mathematics: Linear Algebra F orm ula
Sheet for Electrical GA TE
Matrix Op erations
• Matrix A ddition : F or matrices A and B of the same size.
C = A+B, c
ij
= a
ij
+b
ij
• Matrix Multiplication : F or A (m×n ) and B (n×p ).
C = AB, c
ij
=
n
?
k=1
a
ik
b
kj
• T ransp ose of a Matrix :
(A
T
)
ij
= a
ji
• In v erse of a Matrix : F or a square matrix A (if in v ertible).
AA
-1
= I, A
-1
=
adj(A)
det(A)
where adj(A) is the adjugate matrix, I is the iden tit y matrix.
Determinan ts
• Determinan t of a 2x2 Matrix :
det
[
a b
c d
]
= ad-bc
• Determinan t of a 3x3 Matrix :
det
?
?
a b c
d e f
g h i
?
?
= a(ei-fh)-b(di-fg)+c(dh-eg)
• Prop erties :
det(A
T
) = det(A), det(AB) = det(A)det(B), det(kA) = k
n
det(A) for n×n matrix
• Singular Matrix : det(A) = 0 implies A is non-in v ertible.
1
Page 2


Engineering Mathematics: Linear Algebra F orm ula
Sheet for Electrical GA TE
Matrix Op erations
• Matrix A ddition : F or matrices A and B of the same size.
C = A+B, c
ij
= a
ij
+b
ij
• Matrix Multiplication : F or A (m×n ) and B (n×p ).
C = AB, c
ij
=
n
?
k=1
a
ik
b
kj
• T ransp ose of a Matrix :
(A
T
)
ij
= a
ji
• In v erse of a Matrix : F or a square matrix A (if in v ertible).
AA
-1
= I, A
-1
=
adj(A)
det(A)
where adj(A) is the adjugate matrix, I is the iden tit y matrix.
Determinan ts
• Determinan t of a 2x2 Matrix :
det
[
a b
c d
]
= ad-bc
• Determinan t of a 3x3 Matrix :
det
?
?
a b c
d e f
g h i
?
?
= a(ei-fh)-b(di-fg)+c(dh-eg)
• Prop erties :
det(A
T
) = det(A), det(AB) = det(A)det(B), det(kA) = k
n
det(A) for n×n matrix
• Singular Matrix : det(A) = 0 implies A is non-in v ertible.
1
Eigen v alues and Eigen v ectors
• Characteristic Equation :
det(A-?I) = 0
where ? are eigen v alues.
• Eigen v ector : Non-zero v ector x satisfying:
Ax = ?x
• T race of a Matrix : Sum of eigen v alues equals sum of diagonal elemen ts.
T race(A) =
n
?
i=1
a
ii
=
n
?
i=1
?
i
• Determinan t and Eigen v alues :
det(A) =
n
?
i=1
?
i
Rank and Linear Equations
• Rank of a Matrix : Num b er of linearly indep enden t ro ws or columns.
rank(A)= min(m,n) for m×n matrix
• System of Linear Equations (Ax = b ) :
Consisten t if rank(A) = rank([A|b])
Unique solution if rank(A) = rank([A|b]) = n, else infinite or no solutions
• Solution using In v erse :
x = A
-1
b (if A is in v ertible)
V ector Spaces
• Linear Indep endence : V ectors v
1
,v
2
,...,v
k
are linearly indep enden t if:
c
1
v
1
+c
2
v
2
+···+c
k
v
k
= 0 =? c
1
= c
2
=··· = c
k
= 0
• Basis : Set of linearly indep enden t v ectors that span the v ector space.
Dimension = Num b er of v ectors in basis
• Null Space : Set of v ectors x where Ax = 0 .
Nullit y = n- rank(A)
2
Page 3


Engineering Mathematics: Linear Algebra F orm ula
Sheet for Electrical GA TE
Matrix Op erations
• Matrix A ddition : F or matrices A and B of the same size.
C = A+B, c
ij
= a
ij
+b
ij
• Matrix Multiplication : F or A (m×n ) and B (n×p ).
C = AB, c
ij
=
n
?
k=1
a
ik
b
kj
• T ransp ose of a Matrix :
(A
T
)
ij
= a
ji
• In v erse of a Matrix : F or a square matrix A (if in v ertible).
AA
-1
= I, A
-1
=
adj(A)
det(A)
where adj(A) is the adjugate matrix, I is the iden tit y matrix.
Determinan ts
• Determinan t of a 2x2 Matrix :
det
[
a b
c d
]
= ad-bc
• Determinan t of a 3x3 Matrix :
det
?
?
a b c
d e f
g h i
?
?
= a(ei-fh)-b(di-fg)+c(dh-eg)
• Prop erties :
det(A
T
) = det(A), det(AB) = det(A)det(B), det(kA) = k
n
det(A) for n×n matrix
• Singular Matrix : det(A) = 0 implies A is non-in v ertible.
1
Eigen v alues and Eigen v ectors
• Characteristic Equation :
det(A-?I) = 0
where ? are eigen v alues.
• Eigen v ector : Non-zero v ector x satisfying:
Ax = ?x
• T race of a Matrix : Sum of eigen v alues equals sum of diagonal elemen ts.
T race(A) =
n
?
i=1
a
ii
=
n
?
i=1
?
i
• Determinan t and Eigen v alues :
det(A) =
n
?
i=1
?
i
Rank and Linear Equations
• Rank of a Matrix : Num b er of linearly indep enden t ro ws or columns.
rank(A)= min(m,n) for m×n matrix
• System of Linear Equations (Ax = b ) :
Consisten t if rank(A) = rank([A|b])
Unique solution if rank(A) = rank([A|b]) = n, else infinite or no solutions
• Solution using In v erse :
x = A
-1
b (if A is in v ertible)
V ector Spaces
• Linear Indep endence : V ectors v
1
,v
2
,...,v
k
are linearly indep enden t if:
c
1
v
1
+c
2
v
2
+···+c
k
v
k
= 0 =? c
1
= c
2
=··· = c
k
= 0
• Basis : Set of linearly indep enden t v ectors that span the v ector space.
Dimension = Num b er of v ectors in basis
• Null Space : Set of v ectors x where Ax = 0 .
Nullit y = n- rank(A)
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Key Notes
• Matrix T yp es : Square, diagonal, symmetric (A = A
T
), orthogonal (AA
T
= I ).
• GA TE F o cus : Solv e for determinan ts, eigen v alues, rank, and line ar equations; sim-
plify using ro w reduction.
• Cramer’s R ule : F or n× n system, x
i
=
det(A
i
)
det(A)
, where A
i
is A with i -th column
replaced b y b .
• Ro w Ec helon F orm : Use Gaussian elimination to find rank or solv e equations.
• Eigen v alues : Rep eated ro ots indicate algebraic m ultiplicit y; c hec k geometric m ulti-
plicit y for diagonalization.
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