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Addition
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X + Y = [zij] = [xij + yij]
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Subtraction
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X - Y = [zij] = [xij - yij]
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Multiplication
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X * Y = [zij] = [ xi * yj]
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Constant
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c * X = [zij] = [c * xij]
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Transpose
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XT = [zij] = [xji]
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Tr of Tr
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(XT)T = X
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Tr of Mul
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(XY)T = YT XT != XT YT
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Sym Matrix
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XT = X
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Identity Matrix I
[zii=1, zij=0] |
X I = I X = X
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Inverse
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X X-1 = I = X-1X
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if X-1 exists then X is non singular or invertible
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Inv of Inv
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(X-1)-1 = X
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Inv of Mul
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(XY)-1 = Y-1X-1 != X-1Y-1
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Inv of Tr
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(XT)-1 = (X-1)T
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Determinant
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|A| = n∑
i=1 aij x Det |aij| |
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Determinant is computed over first row of matrix where each element of first row is multiplied by its minor
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minor M
ij is a determinant obtained by deleting the ith row and jth column in which aij lies. Minor of aij is denoted by mij. |
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Cofactor
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A
ij = (-1)i+j mij |
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Adjoint
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adj(A) = (Cofactor)T = (A
ij)T |
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Inverse
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A-1 = adj(A) / |A|
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Two n x 1 vectors are orthogonal if XT Y = 0
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A vector is orthonormal if XTX = ||X2||
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Sq root of ||X|| is length or norm of vector
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{X
1, X2, X3.... Xn) are said to be orthonormal if, each pair is orthogonal and have unit length |
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A sq matrix is orthogonal if XTX= I or XT=X-1
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A is nxn matrix, X is nx1 matrix, λ is a scalar, then
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AX = λX or (A-λI)X = 0 or X = (A-λI)-1
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λ is the eigen value and X is the eigen vector (non zero)
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Since X is non zero, |A-λI| should be 0
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Determinant for [a b] = ad - bc
[c d] |
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If A => symmetric, then eigenvalues => real &
eigenvectors => orthogonal |
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Diagonalization: P => orthogonal matrix, then Z = PTAP, Z is diagonal matrix with eigen values of A
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Given a
1x1 + a2x2 + ...anxn = 0, if a vector [a1, a2, ...an] exists such that |
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a. all a
i are 0, then xi are linearly independent. |
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b. if some a
i!=0 then xi are linearly dependent. |
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If a set of vectors are linearly dependent, then one of them can be written as some combination of others
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A set of two vectors is linearly dependent if and only if one of the vectors is a constant multiple of the other.
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a nxn matrix A is idempotent iff A2 = A
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The identity matrix I is idempotent.
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Let X be an n×k matrix of full rank ,n≥k then H exists as H=X(XTX)−1XT and is idempotent.
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For a nxk matrix say X, the column vectors are [x
1, x2, ...xk] and rank is given by max num of linearly independent vectors. |
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If X is a nxk matrix and r(X) = k, then X is of full rank for n≥k.
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r(X) = r(XT) = r(XTX)
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If X is kxk, then X is non singular iff r(X) = k.
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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).
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The rank of a diagonal matrix is equal to the number of non zero diagonal entries in the matrix.
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r(XY) ≤ r(X) r(Y)
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The trace of a square k×k matrix X is sum of its diagonal entries -
tr(X) = ∑ xii |
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If c is a scalar, tr(cX) =c * tr(X)
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tr(X±Y) =tr(X) ± tr(Y).
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If XY and YX both exist, tr(XY) =tr(YX).
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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
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q = ∑ ∑ a
ijyiyj |
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If yTAy > 0 for all y != 0, yTAy & A are +ve definite
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If yTAy >= 0 for all y != 0, yTAy & A are +ve semidefinite
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y = (y1, y2, . . . , yk)T, z = f(y) then ∂z/∂y = [∂z/∂y
1 ∂z/∂y2 ∂z/∂y3]T |
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z=aTy, ∂z/∂y = a
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z=yTy, ∂z/∂y = 2y
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z=yTAy, ∂z/∂y = Ay+ATy , if A is symmetrix then ∂z/∂y = 2Ay
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Theorem 1
Let A be a symmetric k×k matrix. Then an orthogonal matrix P exists such that PTAP = λ x I, where λ = [λ 1, λ2, .... λn ] are the eigen values of A as nx1 vector |
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Theorem 2
The eigenvalues of idempotent matrices are always either 0 or 1. |
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Theorem 3
If A is a symmetric and idempotent matrix, r(A) =tr(A) |
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Theorem 4
Let A 1,A2,...,Am 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; b. A iAj=AjAi for every pair i,j= 1,2,...,m. |
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Theorem 5
Let A 1,A2,...,Am 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 |
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Theorem 6
Let A 1,A2,...,Am be a collection of symmetric k×k matrices. If the conditions in Theorem 5 are true, then r(∑A i) = ∑r(Ai) |
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Theroem 7
A symmetric matrix A is positive definite if and only if its eigen values are all (strictly) positive |
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Theorem 8
A symmetric matrix A is positive semi-definite if and only if its eigenvalues are all non-negative. |
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