The document discusses different matrix operations and properties that are useful for MIMO communication. It defines a sparse vector as a vector with few non-zero components. It then lists several properties of matrices including properties related to determinants, traces, eigenvalues, and invertible matrices. It also discusses convex and concave functions, defining a convex function as one where a line segment between two points on the graph lies above or on the graph.

1. Usage of Different Matrix Operation for MIMO
Communication
Domain: Wireless Communication Research
Dr. Varun Kumar
Domain: Wireless Communication Research Dr. Varun Kumar (IIIT Surat)Dr. Varun Kumar 1 / 12

2. Outlines
1 Introduction to Null Space of a Matrix
2 Properties of Matrix and Vector
3 Introduction to Convex and Concave Function
4 References
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3. Sparse vector
A vector, which has few components are non-zero and most of them are
zero is called as sparse vector. Ex-
A = [0, 0, 0, 1, 0, 0, 0, 0, 3, 0, 0, 0, 0, 0]
Note:
⇒ Most of the real world physical signal such as
Music
Image
Video
are sparse in some suitable domain like, Wavelet, Fourier.
⇒ This important property is extensively used in signal processing. Ex-
Compressive sensing
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4. Property of matrix
(1) Let A and B are two matrix of size n × p and p × n. As per the
property of matrix.
|In + AB| = |Ip + BA|
where, In is identity matrix of size n × n and |.| denotes the
determinant. Let Z is another matrix of size (n + p) × (n + p)
Z =
In − A
B Ip
Let R2 → R2 − BR1, the modified matrix is
˜Z =
In − A
0 Ip + BA
In this case
|Z| = | ˜Z| = |In||Ip + BA| = |Ip + BA|
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5. Continued–
Let R1 → R1 + BR1, the modified matrix is
ˆZ =
In + AB 0
B Ip
In this case
|Z| = | ˆZ| = |In + AB||Ip| = |In + AB| = | ˜Z|
(2) Trace and determinant of a matrix
Tr(A) =
N
i=1
λi , λi → ith
eigen value
det(|A|) =
N
i=1
λi , λi → ith
eigen value
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6. Continued–
(3) Trace and determinant of a nth exponent of the matrix.
Tr(An
) =
N
i=1
λn
i , λi → ith
eigen value
det(|A|n
) =
N
i=1
λn
i , λi → ith
eigen value
(4) If A is a invertible square matrix of size N × N then A = UΣUH,
where U is a unitary matrix, such that UUH = IN
(5) If A is positive definite matrix, i.e λi > 0 ∀ i = 1, 2, ...N then A−1 is
also a positive definite matrix.
(6) If A and B is invertible matrix, then AB and BA have same eigen
value.
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7. Continued–
(7) If A and B are matrix, where another matrix M exist in such a way
that
A = MBM−1
then A and B are similar matrix.
(8) If ¯x = [x1, x2, ...., xk]T is vector of size 1 × k, then l1 and l2 norms are
related as
l1 ≤
√
kl2
where l1 = x 1= |x1| + ... + |xk| and
l2 = x 2= |x1|2 + ... + |xk|2
(9) Let X is a random vector, such that X = [x1, x2, ...., xn]T . Also
P(X = x1) = P1 and
n
i=1
Pi = 1
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8. Continued–
P =





P1
P2
...
Pn





0
Case 1: Let α < E(X) < β
⇒ E(X) =
n
i=1
P(X = xi )xi =
n
i=1
Pi xi
Here
XT
P ≤ β and − XT
P ≤ −α (1)
Eq (1) shows the intersection of two half space, hence it is convex set.
Case 2: Let P(X ≥ α) ≤ β
Let X = [x1, x2, x3, x4, x5, x6], where x1, x2, x3 < α ⇒ x4, x5, x6 ≥ α
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9. Continued–
⇒ P4 + P5 + P6 ≤ β. In another sense, let ¯a = [0, 0, 0, 1, 1, 1]T then
aT
P ≤ β (2)
Here, eq (2) shows the half space, hence it is also a convex set.
Case 3 : Let E(X2) ≤ β
E(X2
) =
n
i=1
Pi x2
i
Let v = [x2
1 , x2
2 , ....x2
n ]T . Hence,
vT
P ≤ β (3)
Here, eq (3) shows the half space, hence it is also a convex set.
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10. Convex function
Convex function
F(x) is a convex function, iff
F(θx1 + (1 − θ)x2) ≤ θF(x1) + (1 − θ)F(x2)
where x1, x2 ∈ convex set.
Pictorial representation:
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11. Concave function
If F(x) is a convex function, then −F(x) is a concave function. Hence
F(θx1 + (1 − θ)x2) ≥ θF(x1) + (1 − θ)F(x2)
Note:
If a function is represented graphically, i.e, convex downward then it is
called convex function. Ex- f (x) = x2
If a function is represented graphically, i.e, concave downward then it
is called concave function. Ex- f (x) = 2 − x2
If a function has both characteristics, i.e convex and concave
downward then it is neither convex nor concave. Also, it is partially
convex and partially concave. Ex- f (x) = x3
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12. References
J. Navarro, “A very simple proof of the multivariate chebyshev’s inequality,”
Communications in Statistics-Theory and Methods, vol. 45, no. 12, pp. 3458–3463,
2016.
M. I. Jordan and T. M. Mitchell, “Machine learning: Trends, perspectives, and
prospects,” Science, vol. 349, no. 6245, pp. 255–260, 2015.
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