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Overview When Diagonalization Fails An Example Non-Diagonalizable Homogeneous Systems of Linear Differential Equations with Constant Coefficients Bernd Schr¨ oder logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Non-Diagonalizable Homogeneous Systems of Linear Differential Equations with Constant Coefficients

Overview When Diagonalization Fails An Example Non-Diagonalizable Systems of Linear Differential Equations with Constant Coefficients logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Non-Diagonalizable Homogeneous Systems of Linear Differential Equations with Constant Coefficients

Overview When Diagonalization Fails An Example Non-Diagonalizable Systems of Linear Differential Equations with Constant Coefficients 1. These systems are typically written in matrix form as y ′ = A � � y , where A is an n × n matrix and � y is a column vector with n rows. logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Non-Diagonalizable Homogeneous Systems of Linear Differential Equations with Constant Coefficients

Overview When Diagonalization Fails An Example Non-Diagonalizable Systems of Linear Differential Equations with Constant Coefficients 1. These systems are typically written in matrix form as y ′ = A � � y , where A is an n × n matrix and � y is a column vector with n rows. 2. The theory guarantees that there will always be a set of n linearly independent solutions { � y 1 ,...,� y n } . logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Non-Diagonalizable Homogeneous Systems of Linear Differential Equations with Constant Coefficients

Overview When Diagonalization Fails An Example Non-Diagonalizable Systems of Linear Differential Equations with Constant Coefficients 1. These systems are typically written in matrix form as y ′ = A � � y , where A is an n × n matrix and � y is a column vector with n rows. 2. The theory guarantees that there will always be a set of n linearly independent solutions { � y 1 ,...,� y n } . y = c 1 y 1 + ··· + c n 3. Every solution is of the form � � � y n . logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Non-Diagonalizable Homogeneous Systems of Linear Differential Equations with Constant Coefficients

Overview When Diagonalization Fails An Example Non-Diagonalizable Systems of Linear Differential Equations with Constant Coefficients 1. These systems are typically written in matrix form as y ′ = A � � y , where A is an n × n matrix and � y is a column vector with n rows. 2. The theory guarantees that there will always be a set of n linearly independent solutions { � y 1 ,...,� y n } . y = c 1 y 1 + ··· + c n 3. Every solution is of the form � � � y n . x ′ = D 4. If A = Φ D Φ − 1 and � x solves � � x , then A ( Φ x ) � logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Non-Diagonalizable Homogeneous Systems of Linear Differential Equations with Constant Coefficients

Overview When Diagonalization Fails An Example Non-Diagonalizable Systems of Linear Differential Equations with Constant Coefficients 1. These systems are typically written in matrix form as y ′ = A � � y , where A is an n × n matrix and � y is a column vector with n rows. 2. The theory guarantees that there will always be a set of n linearly independent solutions { � y 1 ,...,� y n } . y = c 1 y 1 + ··· + c n 3. Every solution is of the form � � � y n . x ′ = D 4. If A = Φ D Φ − 1 and � x solves � � x , then x ) = Φ D Φ − 1 ( Φ A ( Φ x ) � � logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Non-Diagonalizable Homogeneous Systems of Linear Differential Equations with Constant Coefficients

Overview When Diagonalization Fails An Example Non-Diagonalizable Systems of Linear Differential Equations with Constant Coefficients 1. These systems are typically written in matrix form as y ′ = A � � y , where A is an n × n matrix and � y is a column vector with n rows. 2. The theory guarantees that there will always be a set of n linearly independent solutions { � y 1 ,...,� y n } . y = c 1 y 1 + ··· + c n 3. Every solution is of the form � � � y n . x ′ = D 4. If A = Φ D Φ − 1 and � x solves � � x , then x ) = Φ D Φ − 1 ( Φ A ( Φ x ) = Φ D � � � x logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Non-Diagonalizable Homogeneous Systems of Linear Differential Equations with Constant Coefficients

Overview When Diagonalization Fails An Example Non-Diagonalizable Systems of Linear Differential Equations with Constant Coefficients 1. These systems are typically written in matrix form as y ′ = A � � y , where A is an n × n matrix and � y is a column vector with n rows. 2. The theory guarantees that there will always be a set of n linearly independent solutions { � y 1 ,...,� y n } . y = c 1 y 1 + ··· + c n 3. Every solution is of the form � � � y n . x ′ = D 4. If A = Φ D Φ − 1 and � x solves � � x , then x ) = Φ D Φ − 1 ( Φ x ′ A ( Φ x ) = Φ D x = Φ � � � � logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Non-Diagonalizable Homogeneous Systems of Linear Differential Equations with Constant Coefficients

Overview When Diagonalization Fails An Example Non-Diagonalizable Systems of Linear Differential Equations with Constant Coefficients 1. These systems are typically written in matrix form as y ′ = A � � y , where A is an n × n matrix and � y is a column vector with n rows. 2. The theory guarantees that there will always be a set of n linearly independent solutions { � y 1 ,...,� y n } . y = c 1 y 1 + ··· + c n 3. Every solution is of the form � � � y n . x ′ = D 4. If A = Φ D Φ − 1 and � x solves � � x , then x ′ = ( Φ x ) = Φ D Φ − 1 ( Φ x ) ′ , A ( Φ x ) = Φ D x = Φ � � � � � logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Non-Diagonalizable Homogeneous Systems of Linear Differential Equations with Constant Coefficients

Overview When Diagonalization Fails An Example Non-Diagonalizable Systems of Linear Differential Equations with Constant Coefficients 1. These systems are typically written in matrix form as y ′ = A � � y , where A is an n × n matrix and � y is a column vector with n rows. 2. The theory guarantees that there will always be a set of n linearly independent solutions { � y 1 ,...,� y n } . y = c 1 y 1 + ··· + c n 3. Every solution is of the form � � � y n . x ′ = D 4. If A = Φ D Φ − 1 and � x solves � � x , then x ′ = ( Φ x ) = Φ D Φ − 1 ( Φ x ) ′ , A ( Φ x ) = Φ D x = Φ � � � � � y ′ = A that is, � y = Φ � x solves � � y . logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Non-Diagonalizable Homogeneous Systems of Linear Differential Equations with Constant Coefficients

Overview When Diagonalization Fails An Example Non-Diagonalizable Systems of Linear Differential Equations with Constant Coefficients 1. These systems are typically written in matrix form as y ′ = A � � y , where A is an n × n matrix and � y is a column vector with n rows. 2. The theory guarantees that there will always be a set of n linearly independent solutions { � y 1 ,...,� y n } . y = c 1 y 1 + ··· + c n 3. Every solution is of the form � � � y n . x ′ = D 4. If A = Φ D Φ − 1 and � x solves � � x , then x ′ = ( Φ x ) = Φ D Φ − 1 ( Φ x ) ′ , A ( Φ x ) = Φ D x = Φ � � � � � y ′ = A that is, � y = Φ � x solves � � y . y ′ = A 5. Conversely, every solution of � � y can be obtained as above. logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Non-Diagonalizable Homogeneous Systems of Linear Differential Equations with Constant Coefficients

Overview When Diagonalization Fails An Example Non-Diagonalizable Systems of Linear Differential Equations with Constant Coefficients logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Non-Diagonalizable Homogeneous Systems of Linear Differential Equations with Constant Coefficients

Overview When Diagonalization Fails An Example Non-Diagonalizable Systems of Linear Differential Equations with Constant Coefficients 6. So if we can find a representation A = Φ D Φ − 1 so that x ′ = D y ′ = A � � x is easy to solve, then � � y is also easy to solve. logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Non-Diagonalizable Homogeneous Systems of Linear Differential Equations with Constant Coefficients

Overview When Diagonalization Fails An Example Non-Diagonalizable Systems of Linear Differential Equations with Constant Coefficients 6. So if we can find a representation A = Φ D Φ − 1 so that x ′ = D y ′ = A � � x is easy to solve, then � � y is also easy to solve. 7. Not every matrix is diagonalizable. logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Non-Diagonalizable Homogeneous Systems of Linear Differential Equations with Constant Coefficients