Matakuliah
Tahun
: MATRIX ALGEBRA FOR STATISTICS
: 2009
Trace and Norm
Pertemuan 8
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Defenisi
If A = (a i j) is an n x n matrix, then the sum of
the diagonal elements is called the trace of A
(trace A).
trace A =
= trace AT
Let A be mxn, and let A- be any weak inverse of A.
Trace(A-A) = trace(AA-) = rank A.
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Let α be a scalar and A and B be matrices
Then
a) tr(AT) = trace(A),
b) trace(α A) = α trace(A),
c) trace(A + B) = trace(A) + trace(B),
d) trace(AB) = trace(BA),
e) tr(ATA) = 0 if and only if A = (0).
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If A is real and symmetric, then
If A has real eigenvalues, then
(traceA)2 ≤ Rank A trace(A2).
If A is an n x n real matrix with real eigenvalues
and exactly t of them are nonzero, then
(traceA)2 ≤ t trace(A2)
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Let A be an n x n real matrix
a) If A has real eigenvalues, then (traceA)2 ≤
rank(A) trace(A2).
b) If A is symmetric, (trace A)2 = rank(A) trace(A2)
if and only if there is a
non-negative integer k such that A2 = kA.
c) If A is symmetric, then A2 = A if and only if rank
A = trace A = trace(A2).
d) trace(ATA) ≥ trace(A2), with equality if and only
if A is symmetric
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If A is an n x n real or complex matrix,
then A can be written as A= XY-YX
for some nxn matrices X and Y
if and only if traceA = 0.
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Let A be m x n and B be n x m, both real or
complex matrices.
a) trace(AB) = trace(BA) = trace(AT BT ) =
trace(BT AT )
b) If rn = n and either A or B is symmetric,
then
trace(AB) =
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Matix
Then
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NORM
• Used for measuring distance in vector spaces
and for providing a measure of how close one
matrix is to another.
• Used for finding the best approximation of a
matrix in a given class of matrices by a matrix in
another class (e.g., of lower rank).
• Used for investigating limits of matrix sequences
and series.
• Useful in statistics in the areas of inequalities,
optimization, matrix approximation, matrix
analysis, and numerical analysis.
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A function IIAII defined on all mxm matrices
A, real or complex. is a matrix norm if the
following conditions hold for all mxm
matrices A and B.
a) IIAII ≥0
b) IIAII = 0 if and only if A = (0).
c) IIcAIl = IcIllAIl for any complex scalar c.
d) IIA + BII ≤ IlAIl + IIBII
e) IlABII ≤ IlAII IIBII
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