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Cayley-Hamilton Theorem) The Cayley-llamilton theorem is a powerful theorem in inear algebra that states: every square matrix of real numbers satisfies its own char- cteristic equation. For 2 x 2 matrices this can be seen in the following way. First, we ntroduce the traxe of the matrix tr A = d1,1 + 22 nd the determinant det A = a1,12,a - 41,2a,1- "hen, the chacteristic polynomial is the quadratic given by p(A) = X" – (tr A)A + (det. A). "hen, for any 2 x 2 matrix, if you plug it into this polynomial, you should always get - жго тatrix. h_test Function: Input parameters: • a single 2 x2 matrix for which you would like to test the Cayley Hamilton Theorem Local Variables: • two scalars to represent the trace and determinant. Output parameters: a single 2x 2 which is the result of evaluating the charcteristic polynomial at A (note: think carefully about the constant term) A possible sample case is: > mat_B = c_h_test ([3, 2 ; 2, 0]) at_B =

Cayley-Hamilton Theorem) The Cayley-llamilton theorem is a powerful theorem in inear algebra that states: every square matrix of real numbers satisfies its own char- cteristic equation. For 2 x 2 matrices this can be seen in the following way. First, we ntroduce the traxe of the matrix tr A = d1,1 + 22 nd the determinant det A = a1,12,a - 41,2a,1- "hen, the chacteristic polynomial is the quadratic given by p(A) = X" – (tr A)A + (det. A). "hen, for any 2 x 2 matrix, if you plug it into this polynomial, you should always get - жго тatrix. h_test Function: Input parameters: • a single 2 x2 matrix for which you would like to test the Cayley Hamilton Theorem Local Variables: • two scalars to represent the trace and determinant. Output parameters: a single 2x 2 which is the result of evaluating the charcteristic polynomial at A (note: think carefully about the constant term) A possible sample case is: > mat_B = c_h_test ([3, 2 ; 2, 0]) at_B =

Database System Concepts
Database System Concepts
7th Edition
ISBN:9780078022159
Author:Abraham Silberschatz Professor, Henry F. Korth, S. Sudarshan
Publisher:Abraham Silberschatz Professor, Henry F. Korth, S. Sudarshan
Chapter1: Introduction
Section: Chapter Questions
Problem 1PE
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(Cayley-IHamilton Theorem) The Cayley-Ilamilton theorem is a powerful theorem in Linear algebra that states: every square matrix of real numbers satisfies its own char- acteristic equation. For 2 x 2 matrices this can be seen in the following way. First, we introduce the trace of the matrix tr A 41,1 +22 and the determinant det A = a11a22 - 41,22,1- Then, the chacteristic polynomial is the quadratic given by p(A) = x – (tr A)A + (det. A). Then, for any 2 x 2 matrix, if you plug it into this polynomial, you should always get a zero matrix. c_h_test Function: Input parameters: • a single 2 x2 matrix for which you would like to test the Cayley Ilamilton Theorem Local Variables: • two scalars to represent the trace and determinant Output parameters: a single 2 x2 which is the result of evaluating the charcteristic polynomial at A (note: think carefully about the constant term) A possible sample case is: > mat_B = c_h_test([3, 2 ; 2, 0]) mat B = 0 0
Transcribed Image Text:(Cayley-IHamilton Theorem) The Cayley-Ilamilton theorem is a powerful theorem in Linear algebra that states: every square matrix of real numbers satisfies its own char- acteristic equation. For 2 x 2 matrices this can be seen in the following way. First, we introduce the trace of the matrix tr A 41,1 +22 and the determinant det A = a11a22 - 41,22,1- Then, the chacteristic polynomial is the quadratic given by p(A) = x – (tr A)A + (det. A). Then, for any 2 x 2 matrix, if you plug it into this polynomial, you should always get a zero matrix. c_h_test Function: Input parameters: • a single 2 x2 matrix for which you would like to test the Cayley Ilamilton Theorem Local Variables: • two scalars to represent the trace and determinant Output parameters: a single 2 x2 which is the result of evaluating the charcteristic polynomial at A (note: think carefully about the constant term) A possible sample case is: > mat_B = c_h_test([3, 2 ; 2, 0]) mat B = 0 0
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