Famous Theorems of Mathematics/Algebra/Matrix Theory

< Famous Theorems of Mathematics‎ | Algebra
Organization of a matrix

An m×n matrix M is a function M:A\rightarrow F where A = {1,2...m} × {1,2...n} and F is the field under consideration.

An m×n matrix (read as m by n matrix), is usually written as:


For other related definitions please see this link.

Basic ProofsEdit

1. The set of all m×n matrices forms an abelian group under matrix addition.

Proof: Clearly the sum of two m×n matrices is another m×n matrix. If A and B are two matrices of equal order then working with their (i,j)th entries we have (A+B)_{i,j}=(A_{i,j})+(B_{i,j})=(B_{i,j})+(A_{i,j})=(B+A)_{i,j} which proves A+B = B+A i.e. commutativity. For associativity we proceed similarly so that A + (B + C) = (A + B) + C. Also the m×n matrix with all entries zero is the additive identity. For every matrix A, the matrix -A whose (i,j)th entry is -A_{i,j} is the inverse. So matrices of same order form an abelian group under addition.

2. Scalar Multiplication has the following properties:

1. Left distributivity: (α+β)A = αA+βA.
2. Right distributivity: α(A+B) = αA+αB.
3. Associativity: (αβ)A=α(βA)).
4. 1A = A.
5. 0A= 0.
6. (-1)A = -A.
(0,1,-1,α & β are scalars; A & B are matrices of equal order, 0 is the zero matrix.)
Proof: Start with the left hand side of (1). We will work with the (i,j)th entries. Clearly ((\alpha+\beta)A)_{i,j}=(\alpha+\beta)\cdot A_{i,j}=(\alpha A)_{i,j}+(\beta A)_{i,j} and so (1) is proved. Similarly (2) can be proved. Associativity follows as ((\alpha \beta)A)_{i,j}=(\alpha \beta)\cdot A_{i,j}=\alpha(\beta A_{i,j}). (4), (5) and (6) follow directly from the definition.

3. Matrix multiplication has the following properties:

1. Associativity: A(BC) = (AB)C.
2. Left distributivity: A(B+C) = AB+AC.
3. Right distributivity: (A+B)C = AC+BC.
4. IA = A = AI.
5. α(BC) = (αB)C = B(αC).
(α is a scalar; A, B & C are matrices, I is the identity matrix. A,B,C & I are of orders m×n, n×p, p×r & m×m respectively.)
Proof: We work with the (i,j)th entries and prove (1) only. The proofs for the rest are similar. Now (A(BC))_{i,j}=\sum_{k=1}^n A_{i,k}(BC)_{k,j}=\sum_{k=1}^n A_{i,k}\Big( \sum_{l=1}^p B_{k,l}C_{l,j}\Big)=\sum_{k=1}^n \sum_{l=1}^p A_{i,k}B_{k,l}C_{l,j} and also ((AB)C)_{i,j}=\sum_{l=1}^p (AB)_{i,l}C_{l,j}=\sum_{l=1}^p \Big( \sum_{k=1}^n A_{i,k}B_{k,l}\Big) C_{l,j}=\sum_{k=1}^n \sum_{l=1}^p A_{i,k}B_{k,l}C_{l,j} so that (i,j)th entries on the two sides are equal.

4. Let A and B be m×n matrices. Then:

(i) (kA)^T = kA^T
(ii) (A+B)^T = A^T + B^T
(iii) (AB)^T = B^TA^T
Sketch of Proof: Work with the (i,j) entries as in the previous proofs.

5. Any system of linear equations has either no solution, exactly one solution or infinitely many solutions.

Proof: Suppose a linear system Ax = b has two different solutions given by X and Y. Then let Z = X - Y. Clearly Z is non zero and A(X + kZ) = AX + kAZ = b + k(AX - AY) = b + k(b - b) = b so that X + kZ is a solution to the system for every possible value of k. Since k can assume infinitely many values so clearly we have an infinite number of solutions.

6. Any triangular matrix A satisfying AA^T = A^TA is a diagonal matrix.

Proof: Suppose A is lower triangular. Now the (i,i)th entry of AA^T is given by \sum_{k=1}^n (A_{i,k})(A^T_{k,i})= \sum_{k=1}^n (A_{i,k})(A_{i,k}) = \sum_{k=1}^n (A_{i,k}^2) = \sum_{k=1}^i (A_{i,k}^2). Also the (i,i)th entry of A^TA is given by \sum_{k=1}^n (A^T_{i,k})(A_{k,i}) = \sum_{k=1}^n (A_{k,i})(A_{k,i}) = \sum_{k=1}^n (A_{k,i}^2) = \sum_{k=i}^n (A_{k,i}^2). Now as AA^T = A^TA so \sum_{k=1}^i (A_{i,k}^2) = \sum_{k=i}^n (A_{k,i}^2) and as A_{i,i}^2 can be subtracted from the two sides we are left with \sum_{k=1}^{i-1} (A_{i,k}^2) = \sum_{k=i+1}^n (A_{k,i}^2).

Now if i = 1 then we have 0 = A_{2,1}^2 + A_{3,1}^2 \cdots A_{n,1}^2 which gives us A_{2,1}=A_{3,1}\cdots A_{n,1}=0. Similarly for i =2 we have  0 = A_{2,1}^2 = A_{3,2}^2 + A_{4,2}^2 \cdots A_{n,2}^2 so that A_{3,2}=A_{4,2}\cdots A_{n,2}=0. It is now clear that in this fashion all non diagonal entries of A can be shown to be zero. The proof for an upper triangular matrix is similar.