Diagonalisations and ON-Bases Artem Los (arteml@kth.se) February - PowerPoint PPT Presentation

Diagonalisations and ON-Bases Artem Los (arteml@kth.se) February 17th, 2017 Artem Los (arteml@kth.se) Diagonalisations and ON-Bases February 17th, 2017 1 / 14

Overview What is Diagonalization? 1 Theorems 2 Orthonormal Bases 3 Exam Question 4 Artem Los (arteml@kth.se) Diagonalisations and ON-Bases February 17th, 2017 2 / 14

What is Diagonalization? Artem Los (arteml@kth.se) Diagonalisations and ON-Bases February 17th, 2017 3 / 14

Definition Diagonalization of a matrix Consider matrix A . Let � e n be a basis of its eigenvectors associated e 1 . . . � with eigenvalues λ 1 . . . λ n . Then,   λ 1 . . . 0 . . ... . . AP = P   . .   0 . . . λ n where P is   | | P = e 1 � . . . e n �   | | Artem Los (arteml@kth.se) Diagonalisations and ON-Bases February 17th, 2017 4 / 14

Definition Diagonalization of a matrix Consider matrix A . Let � e n be a basis of its eigenvectors associated e 1 . . . � with eigenvalues λ 1 . . . λ n . Then,   λ 1 . . . 0 . . ... . . AP = P   . .   0 . . . λ n where P is   | | P = e 1 � . . . e n �   | | We can use it to: Identify shape such as an ellipsoid even if was rotated. Compute A 1000 . Artem Los (arteml@kth.se) Diagonalisations and ON-Bases February 17th, 2017 4 / 14

Example from previous lesson (part B) In Eigenvectors (slide 13), we got that T matrix had eigenvalues λ 0 and λ 0 . 5 , with eigenvectors shown below:     0 0 0 0 − 0 . 5 0 0 0 λ 0 : − 2 − 0 . 5 1 . 5 0 λ 0 . 5 : − 2 0 1 . 5 0     0 0 0 0 0 0 − 0 . 5 0 span { ( − 0 . 25 , 1 , 0) , (0 . 75 , 0 , 1) } span { (0 , 1 , 0) } Artem Los (arteml@kth.se) Diagonalisations and ON-Bases February 17th, 2017 5 / 14

Example from previous lesson (part B) In Eigenvectors (slide 13), we got that T matrix had eigenvalues λ 0 and λ 0 . 5 , with eigenvectors shown below:     0 0 0 0 − 0 . 5 0 0 0 λ 0 : − 2 − 0 . 5 1 . 5 0 λ 0 . 5 : − 2 0 1 . 5 0     0 0 0 0 0 0 − 0 . 5 0 span { ( − 0 . 25 , 1 , 0) , (0 . 75 , 0 , 1) } span { (0 , 1 , 0) } Can we now express T as T = PDP − 1 ? Problem. Artem Los (arteml@kth.se) Diagonalisations and ON-Bases February 17th, 2017 5 / 14

Theorems Artem Los (arteml@kth.se) Diagonalisations and ON-Bases February 17th, 2017 6 / 14

Theorems If we can write A as A = PDP − 1 , then we know that: det A = det D Both A and D have the same eigenvalues rank( A ) = rank( D ) Artem Los (arteml@kth.se) Diagonalisations and ON-Bases February 17th, 2017 7 / 14

Theorems If we can write A as A = PDP − 1 , then we know that: det A = det D Both A and D have the same eigenvalues rank( A ) = rank( D ) A matrix A is only diagonalizable if it has n unique eigenvectors. Furthermore, the eigenvectors should span up R n . Artem Los (arteml@kth.se) Diagonalisations and ON-Bases February 17th, 2017 7 / 14

Theorems If we can write A as A = PDP − 1 , then we know that: det A = det D Both A and D have the same eigenvalues rank( A ) = rank( D ) A matrix A is only diagonalizable if it has n unique eigenvectors. Furthermore, the eigenvectors should span up R n . We can also compute high powers much easier: A k = PD k P − 1 Artem Los (arteml@kth.se) Diagonalisations and ON-Bases February 17th, 2017 7 / 14

Example Given P , can A be diagonalized? Problem. � 11 � � 2 � 6 − 1 A = , P = 9 − 4 1 3 Artem Los (arteml@kth.se) Diagonalisations and ON-Bases February 17th, 2017 8 / 14

Example Given P , can A be diagonalized? Problem. � 11 � � 2 � 6 − 1 A = , P = 9 − 4 1 3 Step 1: Use the definition A � x = λ� x , i.e. compute A � x and check that it’s a multiple of � x . Artem Los (arteml@kth.se) Diagonalisations and ON-Bases February 17th, 2017 8 / 14

Example Given P , can A be diagonalized? Problem. � 11 � � 2 � 6 − 1 A = , P = 9 − 4 1 3 Step 1: Use the definition A � x = λ� x , i.e. compute A � x and check that it’s a multiple of � x . � 11 � � 2 � 6 9 − 4 1 Artem Los (arteml@kth.se) Diagonalisations and ON-Bases February 17th, 2017 8 / 14

Example Given P , can A be diagonalized? Problem. � 11 � � 2 � 6 − 1 A = , P = 9 − 4 1 3 Step 1: Use the definition A � x = λ� x , i.e. compute A � x and check that it’s a multiple of � x . � 11 � � 2 � � 28 � 6 = 9 − 4 1 14 Artem Los (arteml@kth.se) Diagonalisations and ON-Bases February 17th, 2017 8 / 14

Example Given P , can A be diagonalized? Problem. � 11 � � 2 � 6 − 1 A = , P = 9 − 4 1 3 Step 1: Use the definition A � x = λ� x , i.e. compute A � x and check that it’s a multiple of � x . � 11 � � 2 � � 28 � � 2 � 6 = = 14 = ⇒ yes 9 − 4 1 14 1 Artem Los (arteml@kth.se) Diagonalisations and ON-Bases February 17th, 2017 8 / 14

Example Given P , can A be diagonalized? Problem. � 11 � � 2 � 6 − 1 A = , P = 9 − 4 1 3 Step 1: Use the definition A � x = λ� x , i.e. compute A � x and check that it’s a multiple of � x . � 11 � � 2 � � 28 � � 2 � 6 = = 14 = ⇒ yes 9 − 4 1 14 1 � 7 � 11 � � − 1 � � � − 1 � 6 = = − 7 = ⇒ yes 9 − 4 3 − 21 3 Artem Los (arteml@kth.se) Diagonalisations and ON-Bases February 17th, 2017 8 / 14

Example Given P , can A be diagonalized? Problem. � 11 � � 2 � 6 − 1 A = , P = 9 − 4 1 3 Step 1: Use the definition A � x = λ� x , i.e. compute A � x and check that it’s a multiple of � x . � 11 � � 2 � � 28 � � 2 � 6 = = 14 = ⇒ yes 9 − 4 1 14 1 � 7 � 11 � � − 1 � � � − 1 � 6 = = − 7 = ⇒ yes 9 − 4 3 − 21 3 Step 2: Find D (i.e. eigenvalues in the diagonal). Artem Los (arteml@kth.se) Diagonalisations and ON-Bases February 17th, 2017 8 / 14

Example Given P , can A be diagonalized? Problem. � 11 � � 2 � 6 − 1 A = , P = 9 − 4 1 3 Step 1: Use the definition A � x = λ� x , i.e. compute A � x and check that it’s a multiple of � x . � 11 � � 2 � � 28 � � 2 � 6 = = 14 = ⇒ yes 9 − 4 1 14 1 � 7 � 11 � � − 1 � � � − 1 � 6 = = − 7 = ⇒ yes 9 − 4 3 − 21 3 Step 2: Find D (i.e. eigenvalues in the diagonal). � 14 � 0 0 − 7 Artem Los (arteml@kth.se) Diagonalisations and ON-Bases February 17th, 2017 8 / 14

Example Given P , can A be diagonalized? Problem. � 11 � � 2 � 6 − 1 A = , P = 9 − 4 1 3 Step 1: Use the definition A � x = λ� x , i.e. compute A � x and check that it’s a multiple of � x . � 11 � � 2 � � 28 � � 2 � 6 = = 14 = ⇒ yes 9 − 4 1 14 1 � 7 � 11 � � − 1 � � � − 1 � 6 = = − 7 = ⇒ yes 9 − 4 3 − 21 3 Step 2: Find D (i.e. eigenvalues in the diagonal). � 14 � 0 0 − 7 Step 3: Confirm that det A = det D . Artem Los (arteml@kth.se) Diagonalisations and ON-Bases February 17th, 2017 8 / 14

O rtho N ormal Bases Artem Los (arteml@kth.se) Diagonalisations and ON-Bases February 17th, 2017 9 / 14

Definition There are two terms worth distinguishing: a , � a · � orthogonal - ’perpendicular’, i.e. � b are orthogonal if � b = 0 Artem Los (arteml@kth.se) Diagonalisations and ON-Bases February 17th, 2017 10 / 14

Definition There are two terms worth distinguishing: a , � a · � orthogonal - ’perpendicular’, i.e. � b are orthogonal if � b = 0 orthonormal - orthogonal and normalized vectors such that a || = || � || � b || = 1 Artem Los (arteml@kth.se) Diagonalisations and ON-Bases February 17th, 2017 10 / 14

Definition There are two terms worth distinguishing: a , � a · � orthogonal - ’perpendicular’, i.e. � b are orthogonal if � b = 0 orthonormal - orthogonal and normalized vectors such that a || = || � || � b || = 1 Caution. Orthogonal matrix means that the set of column and row vectors is an orthonormal set. Artem Los (arteml@kth.se) Diagonalisations and ON-Bases February 17th, 2017 10 / 14

Definition There are two terms worth distinguishing: a , � a · � orthogonal - ’perpendicular’, i.e. � b are orthogonal if � b = 0 orthonormal - orthogonal and normalized vectors such that a || = || � || � b || = 1 Caution. Orthogonal matrix means that the set of column and row vectors is an orthonormal set. Another useful property is that if a matrix is orthogonal, its inverse is the same as its transpose, i.e. A − 1 = A T Artem Los (arteml@kth.se) Diagonalisations and ON-Bases February 17th, 2017 10 / 14

Definition There are two terms worth distinguishing: a , � a · � orthogonal - ’perpendicular’, i.e. � b are orthogonal if � b = 0 orthonormal - orthogonal and normalized vectors such that a || = || � || � b || = 1 Caution. Orthogonal matrix means that the set of column and row vectors is an orthonormal set. Another useful property is that if a matrix is orthogonal, its inverse is the same as its transpose, i.e. A − 1 = A T What does this mean in the context of base change and rotation? Artem Los (arteml@kth.se) Diagonalisations and ON-Bases February 17th, 2017 10 / 14