The Cayley-Hamilton Theorem states that a matrix satisfies its characteristic equation. For example, the characteristic equation of the matrix shown below is as follows. 22 - 61 + 11 = 0 1 -3 and by the theorem you have A = 2 5 A2 - 6A + 11I, = 0 Demonstrate the Cayley-Hamilton Theorem for the matrix A given below. 0 4 1 A = -1 3 1 0 0 -1 STEP 1: Find and expand the characteristic equation. STEP 2: Compute the required powers of A. A : 1 A3 = STEP 3: Write a matrix version of the characteristic equation by replacing 2 with A. (Use I for the 3x3 identity matrix.)

The Cayley-Hamilton Theorem states that a matrix satisfies its characteristic equation. For example, the characteristic equation of the matrix shown below is as follows. 22 - 61 + 11 = 0 1 -3 and by the theorem you have A = 2 5 A2 - 6A + 11I, = 0 Demonstrate the Cayley-Hamilton Theorem for the matrix A given below. 0 4 1 A = -1 3 1 0 0 -1 STEP 1: Find and expand the characteristic equation. STEP 2: Compute the required powers of A. A : 1 A3 = STEP 3: Write a matrix version of the characteristic equation by replacing 2 with A. (Use I for the 3x3 identity matrix.)

Algebra & Trigonometry with Analytic Geometry

13th Edition

ISBN:9781133382119

Author:Swokowski

Publisher:Swokowski

Chapter9: Systems Of Equations And Inequalities

Section9.8: Determinants

Problem 35E

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a Find a symmetric matrix B such that B2=A for A=[2112] b Generalize the result of part a by proving that if A is an nn symmetric matrix with positive eigenvalues, then there exists a symmetric matrix B such that B2=A.
Find an orthogonal matrix P such that PTAP diagonalizes the symmetric matrix A=[1331].
A square matrix is called upper triangular if all of the entries below the main diagonal are zero. Thus, the form of an upper triangular matrix is where the entries marked * are arbitrary. A more formal definition of such a matrix . 29. Prove that the product of two upper triangular matrices is upper triangular.

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