An example of my lecture for undergraduate students "Matrix determinant".

1. Determinant
Alexander Litvinenko
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http://sri-uq.kaust.edu.sa/
Extreme Computing Research Center, KAUST
Alexander Litvinenko Determinant

2. 4*
The structure of the talk
1. Motivation
2. History
3. Geometrical interpretation
4. Properties
5. Definition
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3. 4*
History
1. A notion similar to determinant appears in old-chinese
book ”The Nine Chapters on the Mathematical Art”,
10th-2nd century BCE
2. for 2 × 2 matrices — by Girolamo Cardano in XVI century,
3. Japanese mathematician Seki Takakazu, 1683
4. for higher dimensions by Gottfried W. Leibnitz in 1693,
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4. 4*
Geometrical Interpretation
Matrix is a linear transformation which squish space down or
stretch it out
Determinant measures how the volume of the original do-
main changed.
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5. 4*
Motivation from spatial statistics
Goal: To improve estimation of unknown statistical parameters
in a spatial soil moisture field, Mississippi basin,
[−84.8◦ − 72.9◦] × [32.446◦, 43.4044◦].
Log-likelihood function:
L(θ) = −
n
2
log(2π) −
1
2
log |C(θ)| −
1
2
Z C(θ)−1
Z.
where C = e−
|x−y|
θ is a large matrix and Z available (satellite)
data.
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6. 4*
Properties. Part I.
1. det(I) = |I| = 1, I -identity matrix
2. exchange two rows: reverse the sign of det(A)
3.
ta tb
c d
= t
a b
c d
4.
a + a b + b
c d
=
a b
c d
+
a b
c d
5. two equal rows result in det(A) = 0 (by P2)
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7. 4*
Properties. Part II.
6. Subtract × row-i from row k, then det(A) doesn’t change
(P4, P3, P5)
7. row of zeros, results in det(A) = 0 (P3 with t = 0)
8.
det(A) =
d1 ∗ ∗ ∗
0 d2 ∗ ∗
...
...
... ∗
0 0 0 dn
= d1d2 · ... · dn
(P6, P3, P1)
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8. 4*
Properties III
9. det(A) = 0 exactly when A is singular AND det(A) = 0
when A is non-singular
10. det(A · B) = det(A) · det(B);
learning that
det(A−1) = 1
det(A)
and
det(cA) = cn det(A) (volume!)
11. det(AT ) = det(A)
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9. 4*
Definitions
for n = 2 : det(A) =
a b
c d
=
a 0
c d
+
0 b
c d
=
a 0
c 0
+
a 0
0 d
+
0 b
c 0
+
0 b
0 d
= ad − bc.
for n = 3 : det(A) =
will have 27 terms , many of them will be zeros.
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10. 4*
Formula for 3 × 3 case
for n = 3 : det(A) =
a b c
d e f
g h i
= a(ei − fh) − b(di − fg) + c(dh − eg)
= a ·
e f
h i
− b ·
d f
g i
+ c ·
d e
g h
Here we colored with red: the (1,1)-minor, (1,2)-minor and
(1,3)-minor correspondingly.
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11. 4*
Formula for n × n
If A is n × n matrix, then |A| = n
j=1(−1)(1+j)a1j|A1j|.
Here A1j is (n − 1) × (n − 1) matrix, obtained by deleting the 1st
row and jth column of A.
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12. 4*
Other applications
|Σ| is used in
Kullback-Leibler divergence (KLD) (distance between two
Gaussian distributions):
2DKL = tr(Σ−1
1 Σ0)+(µ1 −µ0)T
Σ−1
1 (µ1 −µ0)−k −ln
|Σ0|
|Σ1|
The entropy of the multivariate normal distribution is
proportional to |Σ|.
Transformation of coordinates.
Multivariate statistics.
Google: eigenvalues of A are the solutions of the
characteristic equation |A − λI| = 0.
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13. 4*
Conclusion
We learned
Geometrical interpretation of the determinant,
Used properties P1-P4 to derive properties P5-P11
Used properties P1-P11 to derive formulas for
determinants
Applications
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14. 4*
Literature
1. youtube lecture N18, of Prof. Gilbert Strang, MIT
2. The determinant. Essence of linear algebra, chapter 5,
https://www.youtube.com/watch?v=Ip3X9LOh2dk&list=
PLZHQObOWTQDPD3MizzM2xVFitgF8hE_ab&index=7
3. Harville, D. A. (1997). Matrix Algebra From a Statistician’s
Perspective. Springer-Verlag.
4. Brookes, M. (2005). ”The Matrix Reference Manual (online)”.
5. Ding, J., Zhou, A. (2007). ”Eigenvalues of rank-one updated
matrices with some applications”. Applied Mathematics Letters. 20
(12): 1223-1226.
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15. 4*
Advanced Properties. Part III.
Let A be n × n matrix
13. det(exp(A)) = exp(tr(A)) or tr(A) = log det(exp(A))
14. for positive A, have tr(I − A−1) ≤ log det(A) ≤ tr(A − I)
15. det
A 0
C D
= det(A) det(D)
16. if A−1 exist det
A B
C D
= det(A) det(D − CA−1B)
17. det(Im + abT ) = 1 + (a, b)
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16. 4*
Advanced Properties. Part IV.
Let A be n × n matrix
18. Matrix determinant Lemma
det(A + abT
) = det(A(I + bT
A−1
a)) = (1 + bT
A−1
a) det(A)
19. U, V is n × m
det(A + UVT
) = det(A(I + VT
A−1
U)) det(A)
20. if W is m × m invertible
det(A + UWVT
) = det(W−1
+ VT
A−1
U)) det(W) det(A).
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17. 4*
Advanced Properties. Part V.
21. det(A) = 0 if and only if rank(A) < n,
22. A−1 exists if and only if det(A) = 0,
23. adding to a row/column a linear combination of any other
rows/columns does not change det(A)
24. if two (or more) rows/colums are linear dependent, then
det(A) = 0
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18. 4*
Formula for 4 × 4 and n × n
|A| =
a b c d
e f g h
i j k
m n o p
= a
f g h
j k
n o p
− e
b c d
j k
n o p
+ i
b c d
f g h
n o p
− m
b c d
f g h
j k
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