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Let A be an n x n matrix. Assume that the inverse matrix of A exists. The inverse matrix can be calculated as follows (Csanky's algorithm). Let p(x) = det(xIn - A). (1) The roots are, by definition, the eigenvalues A₁, A2, ..., An of A. We write p(x): = x + ₁x²-¹+...+ Cn-1x + Cn (2) where cn = (-1)" det(A). Since A is nonsingular we have cn #0 and vice versa. The Cayley-Hamilton theorem states that p(A) = A + C₁ An-¹ + ... + Cn-1A+CnIn = On. (3) Multiplying this equation with A-¹ we obtain 1 A-¹ = −(An-1 + C₁ A²-2 + ... + Cn-1 In). (4) -Cn If we have the coefficients c; we can calculate the inverse matrix A. Let n 8k := Σ№. j=1 Then the s, and c, satisfy the following n x n lower triangular system of linear equations C1 $1 2 010-0 82 $1 3 $3 Sn-1 Sn-2 $1 Sn Since tr(A) = ₁ + ₂ + ... + ₂ = = Sk we find sk for k = 1,..., n. Thus we can solve the linear equation for c;. Finally, using (4) we obtain the inverse matrix of A. Apply Csanky's algorithm to the 3 x 3 matrix 101 A = 100 0 1 1

Let A be an n x n matrix. Assume that the inverse matrix of A exists. The inverse matrix can be calculated as follows (Csanky's algorithm). Let p(x) = det(xIn - A). (1) The roots are, by definition, the eigenvalues A₁, A2, ..., An of A. We write p(x): = x + ₁x²-¹+...+ Cn-1x + Cn (2) where cn = (-1)" det(A). Since A is nonsingular we have cn #0 and vice versa. The Cayley-Hamilton theorem states that p(A) = A + C₁ An-¹ + ... + Cn-1A+CnIn = On. (3) Multiplying this equation with A-¹ we obtain 1 A-¹ = −(An-1 + C₁ A²-2 + ... + Cn-1 In). (4) -Cn If we have the coefficients c; we can calculate the inverse matrix A. Let n 8k := Σ№. j=1 Then the s, and c, satisfy the following n x n lower triangular system of linear equations C1 $1 2 010-0 82 $1 3 $3 Sn-1 Sn-2 $1 Sn Since tr(A) = ₁ + ₂ + ... + ₂ = = Sk we find sk for k = 1,..., n. Thus we can solve the linear equation for c;. Finally, using (4) we obtain the inverse matrix of A. Apply Csanky's algorithm to the 3 x 3 matrix 101 A = 100 0 1 1

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Let A be an n x n matrix. Assume that the inverse matrix of A exists. The inverse matrix can be calculated as follows (Csanky's algorithm). Let p(x) = det(xIn - A). (1) The roots are, by definition, the eigenvalues A₁, A2, An of A. We write p(x) = x² +c₁xn−1+...+ Cn-1x + Cn (2) where cn = (-1) det(A). Since A is nonsingular we have cn #0 and vice versa. The Cayley-Hamilton theorem states that p(A) = A + C₁ An-1 +...+Cn-1A+CnIn = On. (3) Multiplying this equation with A-1 we obtain 1 A-¹ = -(A²−¹ + C₁ A¹−² + ... + Cn-1 In). (4) -Cn If we have the coefficients c; we can calculate the inverse matrix A. Let n Sk := =Σ№. j=1 Then the s; and c; satisfy the following n x n lower triangular system of linear equations 1 0 0 $1 2 0 $2 $1 3 00-0 : Sn-1 Sn-2 $1 Cn Since tr(Ak) = \½ + x₂ + ... + x 2 = Sk we find sk for k = 1,..., n. Thus we can solve the linear equation for c;. Finally, using (4) we obtain the inverse matrix of A. Apply Csanky's algorithm to the 3 x 3 matrix 10 1 00 A = ( 1 0 1 1 588 ..
Transcribed Image Text:Let A be an n x n matrix. Assume that the inverse matrix of A exists. The inverse matrix can be calculated as follows (Csanky's algorithm). Let p(x) = det(xIn - A). (1) The roots are, by definition, the eigenvalues A₁, A2, An of A. We write p(x) = x² +c₁xn−1+...+ Cn-1x + Cn (2) where cn = (-1) det(A). Since A is nonsingular we have cn #0 and vice versa. The Cayley-Hamilton theorem states that p(A) = A + C₁ An-1 +...+Cn-1A+CnIn = On. (3) Multiplying this equation with A-1 we obtain 1 A-¹ = -(A²−¹ + C₁ A¹−² + ... + Cn-1 In). (4) -Cn If we have the coefficients c; we can calculate the inverse matrix A. Let n Sk := =Σ№. j=1 Then the s; and c; satisfy the following n x n lower triangular system of linear equations 1 0 0 $1 2 0 $2 $1 3 00-0 : Sn-1 Sn-2 $1 Cn Since tr(Ak) = \½ + x₂ + ... + x 2 = Sk we find sk for k = 1,..., n. Thus we can solve the linear equation for c;. Finally, using (4) we obtain the inverse matrix of A. Apply Csanky's algorithm to the 3 x 3 matrix 10 1 00 A = ( 1 0 1 1 588 ..
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