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- 1. T- 1-855-694-8886 Email- info@iTutor.com By iTutor.com
- 2. Inverse matrix Review AA-1 = I A-1A=I Necessary for matrix to be square to have unique inverse. If an inverse exists for a square matrix, it is unique (A')-1=(A-1)‘ Solution to A x = d A-1A x* = A-1 d I x* =A-1 d=> x* = A-1 d(solution depends on A-1) Linear independence a problem to get x*
- 3. Inverse of a Matrix: Calculation 100 010 001 | ihg fed cba Process: Append the identity matrix to A. Subtract multiples of the other rows from the first row to reduce the diagonal element to 1. Transform the identity matrix as you go. Theorem: Let A be an invertible (n x n) matrix. Suppose that a sequence of elementary row-operations reduces A to the identity matrix. Then the same sequence of elementary row-operations when applied to the identity matrix yields A-1. When the original matrix is the identity, the identity has become the inverse!
- 4. Determination of the Inverse (Gauss-Jordan Elimination) AX = I I X = K I X = X = A-1 => K = A-1 1) Augmented matrix all A, X and I are (n x n) square matrices X = A-1 Gauss elimination Gauss-Jordan elimination UT: upper triangular further row operations [A I ] [ UT H] [ I K] 2) Transform augmented matrix Wilhelm Jordan (1842– 1899)
- 5. Find A-1 using the Gauss-Jordan method. Gauss-Jordan Elimination: Example 1 211 121 112 A 100211 010121 00 2 1 2 1 2 1 1 100211 010121 001112 |.1 )2/1(1R IA 10 2 1 2 3 2 1 0 01 2 1 2 1 2 3 0 00 2 1 2 1 2 1 1 100211 010121 00 2 1 2 1 2 1 1 .2 )1(&)1( 3121 RR Process: Expand A|I. Start scaling and adding rows to get I|A-1.
- 6. 10 2 1 2 3 2 1 0 0 3 2 3 1 3 1 10 00 2 1 2 1 2 1 1 10 2 1 2 3 2 1 0 01 2 1 2 1 2 3 0 00 2 1 2 1 2 1 1 .3 )3/2(2 A R 1 3 1 3 1 3 4 00 0 3 2 3 1 3 1 10 00 2 1 2 1 2 1 1 10 2 1 2 3 2 1 0 0 3 2 3 1 3 1 10 00 2 1 2 1 2 1 1 .4 )2/1(32R 4 3 4 1 4 1 100 0 3 2 3 1 3 1 10 00 2 1 2 1 2 1 1 1 3 1 3 1 3 4 00 0 3 2 3 1 3 1 10 00 2 1 2 1 2 1 1 .5 )4/3(3R
- 7. 4 3 4 1 4 1 100 4 1 4 3 4 1 010 8 3 8 1 8 5 0 2 1 1 4 3 4 1 4 1 100 0 3 2 3 1 3 1 10 00 2 1 2 1 2 1 1 .6 )2/1(&)3/1( 1323 RR 4 3 4 1 4 1 100 4 1 4 3 4 1 010 8 2 8 2 8 6 001 4 3 4 1 4 1 100 4 1 4 3 4 1 010 8 3 8 1 8 5 0 2 1 1 .7 )2/1(12R 4 3 4 1 4 1 4 1 4 3 4 1 4 1 4 1 4 3 4 3 4 1 4 1 100 4 1 4 3 4 1 010 8 2 8 2 8 6 001 |.8 11 AAI
- 8. Gauss-Jordan Elimination: Example 2 DDI DDI D DDI I I I I I I I XXYX XYXXXXYXXYXXXXRR XYXXYXYY XXYX XXXYXXR XXYXXYXXYXYY XXXYXXRR YYYX XXXYXXR YYYX XYXX XYXX XYXXYXYY YX XX )(0 0 .4 ][where )(0 0 .3 0 0 .2 0 0 0 0 .1 1 1111 11 1 11 ][ 11 11 11 2 1 1 2 11 12 1 1 Partitioned inverse (using the Gauss-Jordan method).
- 9. Trace of a Matrix The trace of an n x n matrix A is defined to be the sum of the elements on the main diagonal of A: trace(A) = tr (A) = Σi aii. where aii is the entry on the ith row and ith column of A. Properties: - tr(A + B) = tr(A) + tr(B) - tr(cA) = c tr(A) - tr(AB) = tr(BA) - tr(ABC) = tr(CAB) (invariant under cyclic permutations.) - tr(A) = tr(AT) - d tr(A) = tr(dA) (differential of trace) - tr(A) = rank(A) when A is idempotent –i.e., A= A2.
- 10. Application: Rank of the Residual Maker We define M, the residual maker, as: M = In - X(X′X)-1 X′ = In - P where X is an nxk matrix, with rank(X)=k Let’s calculate the trace of M: tr(M) = tr(In) - tr(P) = n - k - tr(IT) = n - tr(P) = k Recall tr(ABC) = tr(CAB) => tr(P) = tr(X(X′X)-1 X′) = tr(X′X (X′X)-1) = tr(Ik) = k Since M is an idempotent matrix –i.e., M= M2-, then rank(M) = tr(M) = n - k
- 11. Determinant of a Matrix The determinant is a number associated with any squared matrix. If A is an n x n matrix, the determinant is |A| or det(A). Determinants are used to characterize invertible matrices. A matrix is invertible (non-singular) if and only if it has a non-zero determinant That is, if |A|≠0 → A is invertible. Determinants are used to describe the solution to a system of linear equations with Cramer's rule. Can be found using factorials, pivots, and cofactors! More on this later. Lots of interpretations
- 12. Used for inversion. Example: Inverse of a 2x2 matrix: dc ba A bcadAA )det(|| ac bd bcad A 11 This matrix is called the adjugate of A (or adj(A)). A-1 = adj(A)/|A|
- 13. Determinant of a Matrix (3x3) cegbdiafhcdhbfgaei ihg fed cba ihg fed cba ihg fed cba ihg fed cba Sarrus’ Rule: Sum from left to right. Then, subtract from right to left Note: N! terms
- 14. Determinants: Laplace formula The determinant of a matrix of arbitrary size can be defined by the Leibniz formula or the Laplace formula. Pierre-Simon Laplace (1749–1827). The Laplace formula (or expansion) expresses the determinant |A| as a sum of n determinants of (n-1) × (n-1) sub-matrices of A. There are n2 such expressions, one for each row and column of A. Define the i,j minor Mij (usually written as |Mij|) of A as the determinant of the (n-1) × (n-1) matrix that results from deleting the i-th row and the j-th column of A. Define the Ci,j the cofactor of A as: ||)1( ,, ji ji ji MC
- 15. The cofactor matrix of A -denoted by C-, is defined as the nxn matrix whose (i,j) entry is the (i,j) cofactor of A. The transpose of C is called the adjugate or adjoint of A -adj(A). Theorem (Determinant as a Laplace expansion) Suppose A = [aij] is an nxn matrix and i,j= {1, 2, ...,n}. Then the determinant njnjjjijij ininiiii CaCaCa CaCaCaA ... ...|| 22 2211 Example: 642 010 321 A 0)0(x4)3x2-x61)(1()0(x2 0))2x)1((x3)0(x)1(x2)6x1(x1 x3x2x1|| 131211 CCCA |A| is zero => The matrix is non-singular.
- 16. Determinants: Properties Interchange of rows and columns does not affect |A|. (Corollary, |A| = |A’|.) To any row (column) of A we can add any multiple of any other row (column) without changing |A|. (Corollary, if we transform A into U or L , |A|=|U| = |L|, which is equal to the product of the diagonal element of U or L.) |I| = 1, where I is the identity matrix. |kA| = kn |A|, where k is a scalar. |A| = |A’|. |AB| = |A||B|. |A-1|=1/|A|.
- 17. Notation and Definitions: Summary A (Upper case letters) = matrix b (Lower case letters) = vector n x m = n rows, m columns rank(A) = number of linearly independent vectors of A trace(A) = tr(A) = sum of diagonal elements of A Null matrix = all elements equal to zero. Diagonal matrix = all off-diagonal elements are zero. I = identity matrix (diagonal elements: 1, off-diagonal: 0) |A| = det(A) = determinant of A A-1 = inverse of A A’=AT = Transpose of A |Mij|= Minor of A A=AT => Symmetric matrix AT A =A AT => Normal matrix AT =A-1 => Orthogonal matrix 17
- 18. The End Call us for more Information: www.iTutor.com Visit 1-855-694-8886

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