title

Eigenvectors and eigenvalues | Chapter 14, Essence of linear algebra

description

A visual understanding of eigenvectors, eigenvalues, and the usefulness of an eigenbasis.
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A solution to the puzzle at the end:
https://www.dropbox.com/s/86yddvprfuaafju/Eigenvalue%20puzzle%20solution.pdf?dl=0
Typo: At 12:27, "more that a line full" should be "more than a line full".
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{'title': 'Eigenvectors and eigenvalues | Chapter 14, Essence of linear algebra', 'heatmap': [{'end': 454.12, 'start': 434.413, 'weight': 0.7}, {'end': 713.065, 'start': 682.87, 'weight': 0.933}], 'summary': 'Delves into the challenges of understanding eigenvectors and eigenvalues, emphasizing their significance in preserving vector spans during transformations, and covers their applications in linear algebra, including the use of eigenbasis to express matrix transformations in a different coordinate system.', 'chapters': [{'end': 58.278, 'segs': [{'end': 46.35, 'src': 'embed', 'start': 19.856, 'weight': 0, 'content': [{'end': 25.439, 'text': 'Eigenvectors and eigenvalues is one of those topics that a lot of students find particularly unintuitive.', 'start': 19.856, 'duration': 5.583}, {'end': 33.202, 'text': 'Questions like why are we doing this and what does this actually mean are too often left just floating away in an unanswered sea of computations.', 'start': 26.059, 'duration': 7.143}, {'end': 40.066, 'text': "And as I've put out the videos of the series, a lot of you have commented about looking forward to visualizing this topic in particular.", 'start': 33.903, 'duration': 6.163}, {'end': 46.35, 'text': 'I suspect that the reason for this is not so much that eigenthings are particularly complicated or poorly explained.', 'start': 40.706, 'duration': 5.644}], 'summary': 'Understanding eigenvectors and eigenvalues can be challenging for students, with many seeking visualization of the topic.', 'duration': 26.494, 'max_score': 19.856, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/PFDu9oVAE-g/pics/PFDu9oVAE-g19856.jpg'}], 'start': 19.856, 'title': 'Grasping eigenvectors and eigenvalues', 'summary': 'Delves into the challenges of understanding eigenvectors and eigenvalues, highlighting the significance of visual comprehension and the anticipated visualization of the topic.', 'chapters': [{'end': 58.278, 'start': 19.856, 'title': 'Understanding eigenvectors and eigenvalues', 'summary': "Discusses the challenges students face in understanding eigenvectors and eigenvalues, emphasizing the importance of visual understanding for grasping the concept, as well as the audience's anticipation for visualizing the topic.", 'duration': 38.422, 'highlights': ['The concept of eigenvectors and eigenvalues is often found to be unintuitive for many students, leading to unanswered questions and a need for visualization.', 'Visual understanding is crucial for comprehending eigenvectors and eigenvalues, as it only makes sense in the context of solid visual understanding for preceding topics.', 'Many students are looking forward to visualizing the topic of eigenvectors and eigenvalues, indicating a strong interest and need for improved understanding.']}], 'duration': 38.422, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/PFDu9oVAE-g/pics/PFDu9oVAE-g19856.jpg', 'highlights': ['Visual understanding is crucial for comprehending eigenvectors and eigenvalues, as it only makes sense in the context of solid visual understanding for preceding topics.', 'Many students are looking forward to visualizing the topic of eigenvectors and eigenvalues, indicating a strong interest and need for improved understanding.', 'The concept of eigenvectors and eigenvalues is often found to be unintuitive for many students, leading to unanswered questions and a need for visualization.']}, {'end': 555.657, 'segs': [{'end': 108.442, 'src': 'embed', 'start': 80.334, 'weight': 0, 'content': [{'end': 84.738, 'text': 'To start, consider some linear transformation in two dimensions, like the one shown here.', 'start': 80.334, 'duration': 4.404}, {'end': 94.126, 'text': "It moves the basis, vector i-hat to the coordinates and j-hat to, So it's represented with a matrix whose columns are,", 'start': 85.439, 'duration': 8.687}, {'end': 104.097, 'text': 'and Focus in on what it does to one particular vector and think about the span of that vector, the line passing through its origin and its tip.', 'start': 94.126, 'duration': 9.971}, {'end': 108.442, 'text': 'Most vectors are going to get knocked off their span during the transformation.', 'start': 104.998, 'duration': 3.444}], 'summary': 'Linear transformation in two dimensions alters vector spans.', 'duration': 28.108, 'max_score': 80.334, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/PFDu9oVAE-g/pics/PFDu9oVAE-g80334.jpg'}, {'end': 326.037, 'src': 'embed', 'start': 299.998, 'weight': 1, 'content': [{'end': 308.085, 'text': 'But often a better way to get at the heart of what the linear transformation actually does, less dependent on your particular coordinate system,', 'start': 299.998, 'duration': 8.087}, {'end': 310.767, 'text': 'is to find the eigenvectors and eigenvalues.', 'start': 308.085, 'duration': 2.682}, {'end': 320.374, 'text': "I won't cover the full details on methods for computing eigenvectors and eigenvalues here,", 'start': 315.751, 'duration': 4.623}, {'end': 326.037, 'text': "but I'll try to give an overview of the computational ideas that are most important for a conceptual understanding.", 'start': 320.374, 'duration': 5.663}], 'summary': 'Find eigenvectors and eigenvalues to understand linear transformations.', 'duration': 26.039, 'max_score': 299.998, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/PFDu9oVAE-g/pics/PFDu9oVAE-g299998.jpg'}, {'end': 376.474, 'src': 'embed', 'start': 351.417, 'weight': 4, 'content': [{'end': 359.985, 'text': 'So finding the eigenvectors and their eigenvalues of a matrix A comes down to finding the values of v and lambda that make this expression true.', 'start': 351.417, 'duration': 8.568}, {'end': 367.429, 'text': "It's a little awkward to work with at first, because that left-hand side represents matrix vector multiplication,", 'start': 362.287, 'duration': 5.142}, {'end': 370.551, 'text': 'but the right-hand side here is scalar vector multiplication.', 'start': 367.429, 'duration': 3.122}, {'end': 376.474, 'text': "So let's start by rewriting that right-hand side as some kind of matrix vector multiplication,", 'start': 371.111, 'duration': 5.363}], 'summary': 'Finding eigenvectors and eigenvalues of matrix a involves solving for v and lambda.', 'duration': 25.057, 'max_score': 351.417, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/PFDu9oVAE-g/pics/PFDu9oVAE-g351417.jpg'}, {'end': 465.498, 'src': 'heatmap', 'start': 434.413, 'weight': 0.7, 'content': [{'end': 436.293, 'text': 'And if you watch chapter five and six,', 'start': 434.413, 'duration': 1.88}, {'end': 448.018, 'text': "you'll know that the only way it's possible for the product of a matrix with a non-zero vector to become zero is if the transformation associated with that matrix squishes space into a lower dimension.", 'start': 436.293, 'duration': 11.725}, {'end': 454.12, 'text': 'And that squishification corresponds to a zero determinant for the matrix.', 'start': 449.838, 'duration': 4.282}, {'end': 465.498, 'text': "To be concrete, let's say your matrix A has columns and and think about subtracting off a variable amount, lambda, from each diagonal entry.", 'start': 455.492, 'duration': 10.006}], 'summary': 'Squishing space into a lower dimension leads to zero determinant.', 'duration': 31.085, 'max_score': 434.413, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/PFDu9oVAE-g/pics/PFDu9oVAE-g434413.jpg'}, {'end': 555.657, 'src': 'embed', 'start': 529.471, 'weight': 3, 'content': [{'end': 538.479, 'text': 'which you can read off as saying that the vector V is an eigenvector of A, staying on its own span during the transformation A.', 'start': 529.471, 'duration': 9.008}, {'end': 543.883, 'text': 'In this example, the corresponding eigenvalue is 1, so V would actually just stay fixed in place.', 'start': 538.479, 'duration': 5.404}, {'end': 549.288, 'text': 'Pause and ponder if you need to make sure that that line of reasoning feels good.', 'start': 546.145, 'duration': 3.143}, {'end': 555.657, 'text': 'This is the kind of thing I mentioned in the introduction.', 'start': 553.717, 'duration': 1.94}], 'summary': 'Vector v is an eigenvector of a with eigenvalue 1, remaining fixed during transformation.', 'duration': 26.186, 'max_score': 529.471, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/PFDu9oVAE-g/pics/PFDu9oVAE-g529471.jpg'}], 'start': 59.058, 'title': 'Eigenvectors and eigenvalues', 'summary': 'Provides insights into matrices as linear transformations, emphasizing the concept of eigenvectors and eigenvalues, and their significance in preserving vector spans during transformations. it also explains the computational methods and implications of finding eigenvectors and eigenvalues in linear algebra.', 'chapters': [{'end': 231.335, 'start': 59.058, 'title': 'Understanding matrices and eigenvectors', 'summary': 'Provides insights into matrices as linear transformations, emphasizing the concept of eigenvectors and eigenvalues, and their significance in preserving vector spans during transformations.', 'duration': 172.277, 'highlights': ['Eigenvectors and eigenvalues are crucial in preserving vector spans during transformations', 'Understanding matrices as linear transformations and the impact on vector spans', 'Significance of eigenvectors and eigenvalues in linear systems']}, {'end': 555.657, 'start': 231.335, 'title': 'Understanding eigenvectors and eigenvalues', 'summary': 'Explains the concept of eigenvectors and eigenvalues in linear algebra, highlighting their significance in representing transformations, and elucidating the computational methods and implications of finding eigenvectors and eigenvalues.', 'duration': 324.322, 'highlights': ['The concept of eigenvectors and eigenvalues in linear algebra is crucial for understanding transformations, offering a simpler representation of 3D rotations and other linear transformations, and enabling the identification of the axis of rotation and the corresponding eigenvalue.', "The significance of finding eigenvectors and eigenvalues lies in providing a deeper understanding of linear transformations, independent of the specific coordinate system, and in simplifying the comprehension of a matrix's behavior.", 'The process of finding eigenvectors and eigenvalues involves solving the equation Av = λv, where A is the matrix representing the transformation, v is the eigenvector, and λ is the eigenvalue, which requires identifying the values of v and λ that satisfy the equation.', 'The determination of eigenvectors and eigenvalues involves the manipulation of matrix multiplication and scalar vector multiplication to find the values that satisfy the equation Av = λv, which contributes to a conceptual understanding of the process.', 'The identification of eigenvalues involves manipulating the matrix by subtracting a variable amount, lambda, from each diagonal entry, and finding the value of lambda that makes the determinant of the matrix zero, indicating the transformation squishes space into a lower dimension.', 'When the matrix a minus lambda times the identity squishes space onto a line, it signifies the existence of a non-zero eigenvector v such that a minus lambda times the identity times v equals the zero vector, emphasizing the significance of eigenvectors in staying on their own span during the transformation A.']}], 'duration': 496.599, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/PFDu9oVAE-g/pics/PFDu9oVAE-g59058.jpg', 'highlights': ['Understanding matrices as linear transformations and the impact on vector spans', 'The concept of eigenvectors and eigenvalues in linear algebra is crucial for understanding transformations, offering a simpler representation of 3D rotations and other linear transformations, and enabling the identification of the axis of rotation and the corresponding eigenvalue', "The significance of finding eigenvectors and eigenvalues lies in providing a deeper understanding of linear transformations, independent of the specific coordinate system, and in simplifying the comprehension of a matrix's behavior", 'Eigenvectors and eigenvalues are crucial in preserving vector spans during transformations', 'The process of finding eigenvectors and eigenvalues involves solving the equation Av = λv, where A is the matrix representing the transformation, v is the eigenvector, and λ is the eigenvalue, which requires identifying the values of v and λ that satisfy the equation']}, {'end': 1006.191, 'segs': [{'end': 610.026, 'src': 'embed', 'start': 575.542, 'weight': 3, 'content': [{'end': 582.724, 'text': 'To find if a value lambda is an eigenvalue, subtract it from the diagonals of this matrix and compute the determinant.', 'start': 575.542, 'duration': 7.182}, {'end': 596.672, 'text': 'Doing this, we get a certain quadratic polynomial in lambda, 3 minus lambda times 2 minus lambda.', 'start': 591.004, 'duration': 5.668}, {'end': 602.722, 'text': 'Since lambda can only be an eigenvalue if this determinant happens to be 0,', 'start': 597.819, 'duration': 4.903}, {'end': 610.026, 'text': 'you can conclude that the only possible eigenvalues are lambda equals 2 and lambda equals 3..', 'start': 602.722, 'duration': 7.304}], 'summary': 'Eigenvalues are 2 and 3 for the given matrix.', 'duration': 34.484, 'max_score': 575.542, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/PFDu9oVAE-g/pics/PFDu9oVAE-g575542.jpg'}, {'end': 713.065, 'src': 'heatmap', 'start': 682.87, 'weight': 0.933, 'content': [{'end': 688.937, 'text': 'The only roots of that polynomial are the imaginary numbers, i and negative i.', 'start': 682.87, 'duration': 6.067}, {'end': 693.563, 'text': 'The fact that there are no real number solutions indicates that there are no eigenvectors.', 'start': 688.937, 'duration': 4.626}, {'end': 699.796, 'text': 'Another pretty interesting example worth holding in the back of your mind is a shear.', 'start': 695.874, 'duration': 3.922}, {'end': 713.065, 'text': 'This fixes i-hat in place and moves j-hat 1 over, so its matrix has columns and All of the vectors on the x-axis are eigenvectors with eigenvalue 1,', 'start': 700.497, 'duration': 12.568}], 'summary': 'Polynomial has imaginary roots, no real solutions, and shear example with eigenvectors.', 'duration': 30.195, 'max_score': 682.87, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/PFDu9oVAE-g/pics/PFDu9oVAE-g682870.jpg'}, {'end': 766.299, 'src': 'embed', 'start': 715.626, 'weight': 1, 'content': [{'end': 717.748, 'text': 'In fact, these are the only eigenvectors.', 'start': 715.626, 'duration': 2.122}, {'end': 726.502, 'text': 'When you subtract off lambda from the diagonals and compute the determinant, what you get is 1 minus lambda squared.', 'start': 718.717, 'duration': 7.785}, {'end': 734.946, 'text': 'And the only root of this expression is lambda equals 1.', 'start': 729.503, 'duration': 5.443}, {'end': 741.05, 'text': 'This lines up with what we see geometrically, that all of the eigenvectors have eigenvalue 1.', 'start': 734.946, 'duration': 6.104}, {'end': 747.934, 'text': "Keep in mind though, it's also possible to have just one eigenvalue, but with more than just a line full of eigenvectors.", 'start': 741.05, 'duration': 6.884}, {'end': 753.949, 'text': 'A simple example is a matrix that scales everything by 2.', 'start': 750.006, 'duration': 3.943}, {'end': 760.714, 'text': 'The only eigenvalue is 2, but every vector in the plane gets to be an eigenvector with that eigenvalue.', 'start': 753.949, 'duration': 6.765}, {'end': 766.299, 'text': 'Now is another good time to pause and ponder some of this before I move on to the last topic.', 'start': 762.276, 'duration': 4.023}], 'summary': 'Eigenvectors have only one eigenvalue, which is 1, and can have multiple eigenvectors for that eigenvalue.', 'duration': 50.673, 'max_score': 715.626, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/PFDu9oVAE-g/pics/PFDu9oVAE-g715626.jpg'}, {'end': 848.225, 'src': 'embed', 'start': 825.881, 'weight': 0, 'content': [{'end': 834.41, 'text': 'And the way to interpret this is that all the basis vectors are eigenvectors, with the diagonal entries of this matrix being their eigenvalues.', 'start': 825.881, 'duration': 8.529}, {'end': 840.819, 'text': 'There are a lot of things that make diagonal matrices much nicer to work with.', 'start': 837.236, 'duration': 3.583}, {'end': 848.225, 'text': "One big one is that it's easier to compute what will happen if you multiply this matrix by itself a whole bunch of times.", 'start': 841.96, 'duration': 6.265}], 'summary': 'Diagonal matrices make computations easier with eigenvectors and eigenvalues.', 'duration': 22.344, 'max_score': 825.881, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/PFDu9oVAE-g/pics/PFDu9oVAE-g825881.jpg'}, {'end': 922.836, 'src': 'embed', 'start': 899.546, 'weight': 4, 'content': [{'end': 906.925, 'text': "but I'll go through a super quick reminder here of how to express a transformation currently written in our coordinate system into a different system.", 'start': 899.546, 'duration': 7.379}, {'end': 914.534, 'text': 'Take the coordinates of the vectors that you want to use as a new basis, which in this case means our two eigenvectors.', 'start': 908.413, 'duration': 6.121}, {'end': 919.395, 'text': 'then make those coordinates the columns of a matrix known as the change of basis matrix.', 'start': 914.534, 'duration': 4.861}, {'end': 922.836, 'text': 'When you sandwich the original transformation,', 'start': 920.275, 'duration': 2.561}], 'summary': 'Explains how to express a transformation using eigenvectors in a different system.', 'duration': 23.29, 'max_score': 899.546, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/PFDu9oVAE-g/pics/PFDu9oVAE-g899546.jpg'}, {'end': 988.158, 'src': 'embed', 'start': 962.32, 'weight': 6, 'content': [{'end': 970.849, 'text': 'So if, for example, you needed to compute the 100th power of this matrix, it would be much easier to change to an eigenbasis.', 'start': 962.32, 'duration': 8.529}, {'end': 975.613, 'text': 'compute the 100th power in that system, then convert back to our standard system.', 'start': 970.849, 'duration': 4.764}, {'end': 978.375, 'text': "You can't do this with all transformations.", 'start': 976.674, 'duration': 1.701}, {'end': 982.876, 'text': "A shear, for example, doesn't have enough eigenvectors to span the full space.", 'start': 978.775, 'duration': 4.101}, {'end': 988.158, 'text': 'But if you can find an eigenbasis, it makes matrix operations really lovely.', 'start': 983.696, 'duration': 4.462}], 'summary': 'Finding an eigenbasis simplifies matrix operations, like computing the 100th power.', 'duration': 25.838, 'max_score': 962.32, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/PFDu9oVAE-g/pics/PFDu9oVAE-g962320.jpg'}], 'start': 556.198, 'title': 'Eigenvalues, eigenvectors, and eigenbasis', 'summary': 'Covers eigenvalues, eigenvectors, and their applications in linear algebra, including examples of matrices with corresponding eigenvalues and eigenvectors, as well as the use of eigenbasis to express matrix transformations in a different coordinate system.', 'chapters': [{'end': 872.451, 'start': 556.198, 'title': 'Eigenvalues and eigenvectors', 'summary': 'Discusses eigenvalues, eigenvectors, and their applications in linear algebra, highlighting examples of matrices with corresponding eigenvalues and eigenvectors, such as rotations, shears, and diagonal matrices.', 'duration': 316.253, 'highlights': ['The only possible eigenvalues are lambda equals 2 and lambda equals 3.', 'All of the vectors on the x-axis are eigenvectors with eigenvalue 1.', 'The matrix that scales everything by 2 has the only eigenvalue as 2, with every vector in the plane being an eigenvector.', 'A diagonal matrix contains all basis vectors as eigenvectors, with the diagonal entries being their eigenvalues.']}, {'end': 1006.191, 'start': 876.451, 'title': 'Eigenbasis and matrix transformations', 'summary': 'Discusses using eigenbasis to change coordinate systems, expressing transformations in a different system using a change of basis matrix, and the benefits of working in an eigenbasis for matrix operations.', 'duration': 129.74, 'highlights': ['Using eigenbasis to change coordinate systems and express transformations in a different system using a change of basis matrix.', 'The benefits of working in an eigenbasis for matrix operations, as it guarantees a diagonal matrix with corresponding eigenvalues down the diagonal.', 'The impracticality of using eigenbasis for all transformations, as some transformations like a shear may not have enough eigenvectors to span the full space.']}], 'duration': 449.993, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/PFDu9oVAE-g/pics/PFDu9oVAE-g556198.jpg', 'highlights': ['A diagonal matrix contains all basis vectors as eigenvectors, with the diagonal entries being their eigenvalues.', 'The matrix that scales everything by 2 has the only eigenvalue as 2, with every vector in the plane being an eigenvector.', 'All of the vectors on the x-axis are eigenvectors with eigenvalue 1.', 'The only possible eigenvalues are lambda equals 2 and lambda equals 3.', 'Using eigenbasis to change coordinate systems and express transformations in a different system using a change of basis matrix.', 'The benefits of working in an eigenbasis for matrix operations, as it guarantees a diagonal matrix with corresponding eigenvalues down the diagonal.', 'The impracticality of using eigenbasis for all transformations, as some transformations like a shear may not have enough eigenvectors to span the full space.']}], 'highlights': ['The concept of eigenvectors and eigenvalues is often found to be unintuitive for many students, leading to unanswered questions and a need for visualization.', 'Understanding matrices as linear transformations and the impact on vector spans', 'The process of finding eigenvectors and eigenvalues involves solving the equation Av = λv, where A is the matrix representing the transformation, v is the eigenvector, and λ is the eigenvalue, which requires identifying the values of v and λ that satisfy the equation', "The significance of finding eigenvectors and eigenvalues lies in providing a deeper understanding of linear transformations, independent of the specific coordinate system, and in simplifying the comprehension of a matrix's behavior", 'Eigenvectors and eigenvalues are crucial in preserving vector spans during transformations', 'Using eigenbasis to change coordinate systems and express transformations in a different system using a change of basis matrix', 'A diagonal matrix contains all basis vectors as eigenvectors, with the diagonal entries being their eigenvalues', 'The matrix that scales everything by 2 has the only eigenvalue as 2, with every vector in the plane being an eigenvector', 'All of the vectors on the x-axis are eigenvectors with eigenvalue 1', 'The only possible eigenvalues are lambda equals 2 and lambda equals 3', 'Visual understanding is crucial for comprehending eigenvectors and eigenvalues, as it only makes sense in the context of solid visual understanding for preceding topics', 'Many students are looking forward to visualizing the topic of eigenvectors and eigenvalues, indicating a strong interest and need for improved understanding', 'The benefits of working in an eigenbasis for matrix operations, as it guarantees a diagonal matrix with corresponding eigenvalues down the diagonal', 'The impracticality of using eigenbasis for all transformations, as some transformations like a shear may not have enough eigenvectors to span the full space']}