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Linear Algebra Brahmastra
Prassanna Nand Jha
1 Some useful facts :
- Trace(AB) = Trace(BA)
- Diagonal entries of a hermitian matrix are real.
- Diagonal entries of a skew hermitian matrix are zero or purely imaginary.
- Diagonal entries of a skew symmetric matrix are all zero.
- We can have invertible AB without having invertible A, B and BA.
- adj(AB) = adj(B) · adj(A).
- If A ∈ Mn(F), where F is a field, then A · adj(A) = A · adj(A) = det(A) · In×n.
- det(adj(· · · (adj(A)))
| {z }
k times
) = det(A)
(n−1)
k
.
- If A ∈ M n (R) and T race(A) = 0, then ∃ B, C ∈ M n (R) such that A = BC − CB.
- If W (f 1 , f 2 , · · · , fn) =
f 1 f
′
1
· · · f
(n−1)
1
f 2 f
′
2
· · · f
(n−1)
2
. . .
f n f
′
n
· · · f
(n−1)
n
and W (x) ̸= 0 ∀ x ∈ [a, b], then f 1 , f 2 , · · · , fn are L.I.
2 Rank of a matrix:
- ρ(A) = 0 ⇐⇒ A = O and ρ(A) = n ⇐⇒ A is invertible.
- ρ(A) = ρ(A) = ρ(A
T ) = ρ(A
θ ) = ρ(A
θ A) = ρ(AA
θ ).
- If A m×n and B n×m , then ρ(AB) = ρ(BA).
- Rank of a real skew-symmetric matrix is always even.
- Rank of an idempotent complex/real matrix is always equal to its trace.
- ρ(A) + nullity(A) = n, number of columns.
- If A m×n
,B
m×m and C n×n are invertible, then ρ(A) = ρ(BA) = ρ(AC) = ρ(BAC).
- If Bn×p, then ρ(A) + ρ(B) − n ≤ ρ(AB) ≤ min(ρ(A), ρ(B)).
- ρ(A + B) ≤ ρ(A) + ρ(B).
- If B is obtained by changing k elements of A, then |ρ(A) − ρ(B)| ≤ k.
- If A ∈ M n (R), then ρ(adj(A)) =
n : ρ(A) = n
1 : ρ(A) = n − 1
0 : otherwise
So, if ρ(A) ≤ n − 2, then adj(A) = O.
3 System of equations :
Let AX = b denote the matrix representation of a system of equations. Then
- ρ(A : b) ̸= ρ(A) ⇐⇒ ρ(A : b) = ρ(A) + 1 ⇐⇒ No solutions.
- ρ(A : b) = ρ(A) ⇐⇒ Has solutions.
- ρ(A : b) = ρ(A) = number of columns of A ⇐⇒ Unique solution.
- ρ(A : b) = ρ(A) < number of columns of A ⇐⇒ Infinitely many solutions.
- b = O =⇒ Has solutions.
- ρ(A) = number of columns of A ⇐⇒ X = O is the only solution.
- ρ(A) < number of columns of A ⇐⇒ Infinitely many solutions.
Besides, Number of linearly independent solutions = number of columns of A - ρ(A).
- b ̸= O =⇒ can or cannot have solutions.
- ρ(A : b) ̸= ρ(A) ⇐⇒ ρ(A : b) = ρ(A) + 1 ⇐⇒ No solutions.
- ρ(A : b) = ρ(A) ⇐⇒ Has solutions.
∗ ρ(A : b) = ρ(A) = number of columns of A ⇐⇒ Unique solution X = A
− 1 b
∗ ρ(A : b) = ρ(A) < number of columns of A ⇐⇒ Infinitely many solutions.
∗ Number of linearly independent solutions = number of columns of A - ρ(A) + 1.
- If A ∈ M n (R), then
- det(A) ̸= 0 =⇒ Unique solution.
- det(A) = 0 =⇒ Infinitely many solutions.
4 Characteristic and minimal polynomials
- For A ∈ M n (R), characteristic polynomial χ A (λ) = λ
n
n− 1
- For every p(t) = c 0 + c 1 t + · · · + c n− 1 t
n− 1
n , we have a matrix
C(p) =
0 0 · · · 0 −c 0
1 0 · · · 0 −c 1
0 1 · · · 0 −c 2
0 0 · · · 1 −c n− 1
such that χ A (t) = m A (t) = p(t).
- If λ is an eigenvalue of A with respect to the eigenvector v, then p(λ) is an eigenvalue
of p(A) for any polynomial p(x) with respect to the eigenvector v.
- A is invertible. ⇐⇒ The constant term of m A (λ) is nonzero.
- If m < n, A m×n and B n×m , then χ BA (λ) = λ
n−m χ AB (λ)
- For A, B ∈ Mn(F), χAB (λ) = χBA(λ) but mAB (λ) ̸= mBA(λ).
But if A or B is invertible, then even m AB (λ) = m BA (λ).
- det(A) = 0 ⇐⇒ 0 ∈ Spec(A).
- If ρ(A n×n ) = 1, then m A (λ) = 2.
6 Similar matrices
A ∼ B ⇐⇒ ∃ P such that B = P
− 1 AP. If A ∼ B, then
- p(A) ∼ p(B), where p(x) is any polynomial.
- Their eigenvalues are same and with the same multiplicities.
- If v is an eigenvector of A corresponding to λ, then P
− 1 v is the eigenvector
of B corresponding to λ.
- χA(λ) = χB (λ)
- m A (λ) = m B (λ)
- T race(A) = T race(B)
- Det(A) = Det(B)
- ρ(A) = ρ(B)
Remark : Even if χ A (λ) = χ B (λ) and m A (λ) = m B (λ), we can’t claim that A ∼ B.
7 Diagonalization
A is diagnolizable ⇐⇒ A ∼ D, a diagonal matrix.
- If A ∈ Mn(F) and spec(A) ∈ F, the following are equivalent.
- A is diagonalizable over F.
- If v 1 , v 2 , · · · v n are the eigenvectors of A and P = [v 1 v 2 v 3 · · · v n ], then
P
− 1 AP =
λ 1
0 λ 2
0 0 · · · λ n
∗ A.M.(λ) = multiplicity of x − λ in χ A (λ).
∗ G.M.(λ) = dim(Ker(A − λI)) = Number of L.I. eigenvectors corresponding to λ.
- m A (λ) splits into distinct linear factors on F.
- A has n linearly independent eigenvectors.
X
λ∈Spec(A)
G.M.(λ) = n
- If A n×n has n distinct eigenvalues over F, then A is diagnolizable over F.
- Every real symmetric matrix is diagnolizable over R.
Besides, ∃ an orthogonal matrix P such that
P
− 1 AP = P
T AP =
λ 1 0 · · · 0
0 λ 2 · · · 0
0 0 · · · λ n
- A real skew-symmetric matrix is diagonalizable over R ⇐⇒ It is the zero matrix.
- A nilpotent matrix is diagonalizable over R ⇐⇒ It is the zero matrix.
8 Vector Spaces
- All vector spaces are abelian groups under vector addition. Hence, must contain zero.
- If F is a subfield of E, then E is a vector space over F.
- If F is a proper subfield of E, then F is NOT a vector space over E.
- If F is a subfield of E and V is a vector space over E, then V is a vector space over F.
- If X ̸= 0, then R
X is a vector space over R.
- Set of symmetric matrices is a subspace of M n (R) but
set of hermitian matrices is NOT a subspace of M n
(C).
- Set of skew-symmetric matrices is a subspace of M n (R) but
set of skew-hermitian matrices is NOT a subspace of M n
(C).
- Set of orthogonal matrices is NOT a subspace of M n (R) and
set of unitary matrices is NOT a subspace of M n
(C).
- Neither the set of nilpotent matrices, nor the set of idempotent matrices are subspaces of Mn(R) or Mn(C).
- Set of diagonalizable matrices over R is NOT a subspace of Mn(R).
- In M n (C), the set {A ∈ M n (C) : trace(A) = 0} is a vector space and
{A ∈ M
n (C) : trace(A) = 0} = span({A ∈ M n (C) : A is nilpotent})
- If S = {(α 1 , α 2 , α 3 ), (β 1 , β 2 , β 3
)} ⊆ R
3 (R), then
(a, b, c) ∈ L(S) ⇐⇒
a b c
α 1 α 2 α 3
β 1 β 2 β 3
- Arbitrary intersection of subspaces is a subspace.
- If W 1 and W 2 are subspace of V , then W 1
∪ W
2 is a subspace of V ⇐⇒ W 1
⊆ W
2 or W 2
⊆ W
1
, S
2 ⊆ V , then L(S 1
∪ S
2
) = L(S
1
) + L(S
2
- If W 1 and W 2 are subspaces of V , then L(W 1
∪ W
2
) = W
1
+ W
2
- Dimension of the vector space R over field Q is uncountable.
- If A ∈ M n (R) and W A
= L
R
{A
k : k = 0, 1 , 2 , · · · }, then dim(W A ) = deg(m A (λ)) and
W
A
= L
R
{I, A, A
2 , · · · , A
m− 1 }, where m = deg(m A (λ))
- If W 1 and W 2 are subspaces a finite dimensional vector space V , then
dim(W 1
+ W
2 ) = dim(W 1 ) + dim(W 2 ) − dim(W 1
∩ W
2
- The only proper subspaces of R
2 are lines crossing through (0, 0).
- Any vector space over an infinite field CANNOT be written as a finite union of proper subspaces of V.
