Matrices, Determinants and Linear Systems

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1 September 21, 2014

2 Matrices A matrix A m n is an array of numbers in rows and columns a 11 a 12 a 1n r 1 a 21 a 22 a 2n r a m1 a m2 a mn r m c 1 c 2 c n We say that the dimension of A is m n (we also could say m-by-n). If m = n, we say that A is square, otherwise we say that it is rectangular. Recommended videos to follow the class at Khan Academy (click)

3 Matrices Main diagonal. If A is a square matrix of dimension n, the elements a ii, i = 1,...,n form the main diagonal of the matrix; the sum of these elements is called the trace of the matrix. Transpose of a matrix. it is the matrix obtained when interchanging rows and columns.

4 Generalities on Matrices Special shapes: Khan Academy Row matrix, column matrix. Diagonal matrix. Identity matrix I. Null matrix O. Triangular matrices (upper, lower). Symmetric matrix, skew-symmetric matrix. Diagonal by blocks matrix.

5 Matrices Operations: Khan Academy 1. Addition. Properties: Commutative: A+B = B +A Associative: A+(B +C) = (A+B)+C Neutral element: null matrix. A+O = A Inverse element: negative of a matrix. A+( A) = O 2. Multiplication by a number. Properties: λ (A+B) = λ A+λ B, (λ+µ) A = λ A+µ A λ (µ A) = (λ µ) A 1 A = A.

6 Matrices Operations: Khan Academy 3. Multiplication of two matrices. Properties: Not commutative, in general. i.e. A B B A. Associative: A (B C) = (A B) C Neutral element for square matrices: identity matrix. A I = I A = A Inverse element for certain square matrices: inverse matrix. (A B) T = B T A T. Obs: The matrices A and B are said to commute if A B = B A. A is said regular, invertible if it has inverse, otherwise is said singular.

7 Matrices Inverse of a matrix: given an square matrix A, A 1 (its inverse) is the matrix, if it exists, fulfilling A A 1 = A 1 A = I A 1 does not always exist. A characterization of its existence can be achieved by using determinants, or the notion of rank. (A 1 ) T = (A T ) 1. (A B) 1 = B 1 A 1 Two ways for computing it: determinants or Gauss-Jordan method (see later).

8 Determinants Given a square matrix A, the determinant of A, denoted by A, or det(a), is a number associated with A. A is defined first for 2 2 matrices Khan Academy. For matrices of higher order, in principle it can be computed by expanding along any row/column. For example, if A is 3 3, then we can expand along the first row (we might also choose any other row or column) a 11 a 12 a 13 a 21 a 22 a 23 a 31 a 32 a 33 = a 11 A 11 +a 12 A 12 +a 13 A 13 where A ij denotes the cofactor of the entry a ij (i.e. the signed minor of a ij ). Also, for 3 3 matrices, Sarrus rule may be useful.

9 Determinants Basic properties: 1. A = A t 2. If A and B are square matrices of the same order, then A B = A B. 3. If all the elements in a row (or column) admit a same factor, then that number can be taken out of the determinant. 4. If we interchange two rows (or columns), the determinant changes sign.

10 Determinants Basic properties: 5. If A has a row or a column of 0 s, then det(a) = If A has two rows (or columns) which are either equal or proportional, then det(a) = 0. The value is also 0 if there is some row (column) which is a linear combination of others. 7. The value of the determinant does not change if we add to a row (or column) other rows (or columns) multiplied by numbers. This property is essential for efficiently computing determinants. Efficient computation of determinants: Khan Academy

11 Determinants Computation of the inverse of an square matrix A. The inverse A 1 exists if and only if A = 0. A 1 = 1 A AdjT (A) = 1 A Adj(AT ), where Adj(A) is the cofactor matrix, i.e. for each i, j, the corresponding element is the cofactor of a ij. Alternative: Gauss method. Khan Academy

12 Rank of a Matrix We say that a certain row r (similarly for columns) is a linear combination of the rows r i1,...,r is if r can be obtained from these rows by means of an expression like α 1 r i1 + +α s r is for certain numbers α 1,...,α s, which are called the coefficients of the linear combination. We say that certain rows (similarly for columns) are linearly independent, if none of them can be obtained as a linear combination of the rest. Otherwise, we say that they are linearly dependent. Question: When are two rows (resp. two columns) linearly dependent?

13 Rank of a Matrix Definition The rank of a matrix A, rank(a), is the maximum number of rows (or columns) which are linearly independent. An equivalent definition of rank, in terms of determinants. A minor, in a matrix A, is any determinant that you can get by eliminating some rows and/or columns. Then one may see that rank(a) is the maximum order of the non-zero minors of A. (explanation: Khan Academy)

14 Rank of a Matrix Some observations/properties: We say that a square matrix of orden n has full rank (or is regular), if rank(a) = n. It can be proven that this happens if and only if the determinant of A is different from 0 (therefore, if and only if A is invertible). If A is square and has not full rank, it is called singular; such a matrix has no inverse. The rank by rows coincides with the rank by columns. rank(a) = rank(a T ). If the dimension of A is m n, then rank(a) min(m,n). When we compute the rank, we find rows/columns which are linearly independent!

15 Rank of a Matrix Some rules for computing rank(a): A matrix has rank 0 if and only if all its elements are 0, i.e., A = O. A row/column of 0 s does not count for determining the rank of A. Similarly, a row/colum which is clearly a multiple of another row/colum, or a linear combination of rows/columns, does not count, either. The rank does not change if we perform elementary operations on A (swapping rows/columns, multiplying a row/column by a number, add a linear combination of rows/column to another row/colum). From a practical point of view, the computation of the rank can be done either using determinants or by means of Gauss method. Khan Academy

16 Linear Systems: Definitions A System of Linear Equations is a set of equations of the type a 11 x 1 +a 12 x 2 + a 1n x n = b 1 a 21 x 1 +a 22 x 2 + a 2n x n = b 2. a m1 x 1 +a m2 x 2 + a mn x n.. = b m x i s: unknowns a ij s: coefficients b j s: constant terms

17 Linear Systems: Definitions The system can be written in matrix form in the following way: a 11 a 12 a 1n x 1 b 1 a 21 a 22 a 2n x = b 2. a m1 a m2 a mn x n b m Abbreviately, A x = b A: Coefficients matrix. x: vector of unknowns b: vector of constant terms. Question: why is the above equality true?

18 Classification of Linear Systems Classification of Linear Systems: linear systems can be 1 Inconsistent, if it has no solution. 2 Consistent, if it has some solution (i.e. it is solvable). In this case, it can have A Unique solution, or Infinitely many solutions.

19 Classification of Linear Systems In order to classify a given linear system, we use the augmented matrix, and Rouche-Fröbenius Theorem. The augmented matrix, that we denote by B, is a 11 a 12 a 1n b 1 a 21 a 22 a 2n b 2 B = a m1 a m2 a mn b m

20 Classification of Linear Systems Theorem (Rouche-Fröbenius Theorem) Let A x = b be a linear system of m equations with n unknowns, and let B denote the augmented matrix of the system. Then the system is consistent if and only if rank(a) = rank(b); furthermore, the system has a unique solution if rank(a) = rank(b) = n, and infinitely many solutions if rank(a) = rank(b) < n. When rank(a) = rank(b) = n, the difference n rank(a) is in fact the number of degrees of freedom of the system, i.e. the number of parameters the solutions depend on.

21 Solving Linear Systems Two possibilities: 1 Cramer s Method: uses determinants and must be applied on a Cramer s system (i.e. a system where the coefficients matrix has full rank). It is not efficient for big systems. 2 Gauss and Gauss-Jordan Method: does not require to compute determinants, but just simple operations with rows/columns. Efficient for big systems. In both cases, Khan Academy

22 Homogeneous Linear Systems These are those linear systems where the constant terms are all 0: a 11 x 1 +a 12 x 2 + a 1n x n = 0 a 21 x 1 +a 22 x 2 + a 2n x n = 0... a m1 x 1 +a m2 x 2 + a mn x n = 0 Always consistent (why?) The interesting question is whether it has other solutions, up to the trivial one (in such a case it has infinitely many!) This happens only if rank(a) < n. If A is square, this is equivalent to A = 0.

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