Matrix Algebra Flashcards
diagonal matrix
if a matrix conatains zeros in all off-diagonal positions
identify matrix
a diagonal matrix with a 1 in each diagonal position
upper triangular matrix
a square matrix with zeros below the diagonal
j
A vector 1
J
a square matrix of 1’s
o
a vector of zeros
O
a matrix of zeros
(A + B)’ =
A’ + B’
(A - B)’ =
A’ - B’
conformable
the number of colums in A must be the same as the number of rows in B
A (B + C) =
AB + AC
A (B - C) =
AB - AC
(A + B) C =
AC + BC
(A - B) C =
AC - BC
(A - B) (C - D)
(A - B)C - (A - B) D
= AC - BC - AD + BD
ABC =
A (BC) = (AB) C
ABC + ADC =
A (B + D) C
X’X - X’AX =
X’(X - AX) = X’ (1 - A )X
a’a
dot product
aa’
matrix product
j’j =
n
jj’ =
J
a2’ =
(a21 a22 a23 a24)
Partitioned Matrices
Partition a matrix into submatrices
Linearly dependent
if constants c1, c2, …., cn (not all zero) can be found such that
c1a1 + c2a2 + … + can = 0
If A is n times p, the maximum possible rank of A is the smaller of n and p
Full rank (full row rank or full column rank)
A’
transpose by interchanging rows and columns
