X + Y = [zij] = [xij + yij]
X - Y = [zij] = [xij - yij]
X * Y = [zij] = [ xi * yj]
c * X = [zij] = [c * xij]
Transpose & Identity
XT = [zij] = [xji]
Tr of Tr
(XT)T = X
Tr of Mul
(XY)T = YT XT != XT YT
XT = X
Identity Matrix I
X I = I X = X
X X-1 = I = X-1X
if X-1 exists then X is non singular or invertible
Inv of Inv
(X-1)-1 = X
Inv of Mul
(XY)-1 = Y-1X-1 != X-1Y-1
Inv of Tr
(XT)-1 = (X-1)T
|A| = n∑
j x Det |a
Determinant is computed over first row of matrix where each element of first row is multiplied by its minor
j is a determinant obtained by deleting the ith row and jth column in which a
j lies. Minor of a
j is denoted by m
j = (-1)i+j m
adj(A) = (Cofactor)T = (A
A-1 = adj(A) / |A|
Two n x 1 vectors are orthogonal if XT Y = 0
A vector is orthonormal if XTX = ||X2||
Sq root of ||X|| is length or norm of vector
n) are said to be orthonormal if, each pair is orthogonal and have unit length
A sq matrix is orthogonal if XTX= I or XT=X-1
Eigen Values & Eigen Vectors
A is nxn matrix, X is nx1 matrix, λ is a scalar, then
AX = λX or (A-λI)X = 0 or X = (A-λI)-1
λ is the eigen value and X is the eigen vector (non zero)
Since X is non zero, |A-λI| should be 0
Determinant for [a b] = ad - bc
If A => symmetric, then eigenvalues => real &
eigenvectors => orthogonal
Diagonalization: P => orthogonal matrix, then Z = PTAP, Z is diagonal matrix with eigen values of A
1 + a
2 + ...a
n = 0, if a vector [a
n] exists such that
a. all a
i are 0, then x
i are linearly independent.
b. if some a
i!=0 then x
i are linearly dependent.
If a set of vectors are linearly dependent, then one of them can be written as some combination of others
A set of two vectors is linearly dependent if and only if one of the vectors is a constant multiple of the other.
a nxn matrix A is idempotent iff A2 = A
The identity matrix I is idempotent.
Let X be an n×k matrix of full rank ,n≥k then H exists as H=X(XTX)−1XT and is idempotent.
For a nxk matrix say X, the column vectors are [x
k] and rank is given by max num of linearly independent vectors.
If X is a nxk matrix and r(X) = k, then X is of full rank for n≥k.
r(X) = r(XT) = r(XTX)
If X is kxk, then X is non singular iff r(X) = k.
If X is n×k, P is n×n and non-singular, and Q is k×k and nonsingular, then r(X) =r(PX) =r(XQ).
The rank of a diagonal matrix is equal to the number of non zero diagonal entries in the matrix.
r(XY) ≤ r(X) r(Y)
The trace of a square k×k matrix X is sum of its diagonal entries -
tr(X) = ∑ xii
If c is a scalar, tr(cX) =c * tr(X)
tr(X±Y) =tr(X) ± tr(Y).
If XY and YX both exist, tr(XY) =tr(YX).
A be a k × k, y be k × 1 vector containing variables q = yT Ay is called a quadratic form in y, A is called the matrix of the quadratic form
q = ∑ ∑ a
If yTAy > 0 for all y != 0, yTAy & A are +ve definite
If yTAy >= 0 for all y != 0, yTAy & A are +ve semidefinite
y = (y1, y2, . . . , yk)T, z = f(y) then ∂z/∂y = [∂z/∂y
z=aTy, ∂z/∂y = a
z=yTy, ∂z/∂y = 2y
z=yTAy, ∂z/∂y = Ay+ATy , if A is symmetrix then ∂z/∂y = 2Ay
Let A be a symmetric k×k matrix. Then an orthogonal matrix P exists such that PTAP = λ x I, where λ = [λ
2, .... λ
n ] are the eigen values of A as nx1 vector
The eigenvalues of idempotent matrices are always either 0 or 1.
If A is a symmetric and idempotent matrix, r(A) =tr(A)
m be a collection of symmetric k×k matrices. Then the following are equivalent:
a. There exists an orthogonal matrix P such that PTA
iP is diagonal for all i= 1,2,...,m;
i for every pair i,j= 1,2,...,m.
m be a collection of symmetric k×k matrices. Then any two of the following conditions implies the third:
a. All Ai, i= 1,2,...,m are idempotent;
b. ∑ Ai is idempotent;
c. AiAj= 0for i6=j
m be a collection of symmetric k×k matrices. If the conditions in Theorem 5 are true, then
i) = ∑r(A
A symmetric matrix A is positive definite if and only if its eigen values are all (strictly) positive
A symmetric matrix A is positive semi-definite if and only if its eigenvalues are all non-negative.