Welcome to MIT 18.06: Linear Algebra! The Spring 2023 course information, materials, and links are recorded below. Course materials for previous semesters are archived in the other branches of this repository. You can dive in right away by reading this introduction to the course by Professor Strang.
NOTICE OF VIDEO RECORDING: The Spring 2023 lectures for 18.06 will be recorded for OCW. See the notice of recording for more information.
Catalog Description: Basic subject on matrix theory and linear algebra, emphasizing topics useful in other disciplines, including systems of equations, vector spaces, determinants, eigenvalues, singular value decomposition, and positive definite matrices. Applications to least-squares approximations, stability of differential equations, networks, Fourier transforms, and Markov processes. Uses linear algebra software. Compared with 18.700, more emphasis on matrix algorithms and many applications.
Lectures: Monday, Wednesday, and Friday at 11am in 26-100.
Instructors: Prof. Gilbert Strang and Dr. Andrew Horning.
Textbook: Introduction to Linear Algebra: 6th Edition. Professor Strang will explain more about this new 6th edition in class (it is not yet on Amazon). It now ends with two chapters on deep learning. Professor Strang plans to make the textbook available for students to purchase at a discount. Here again is a link to the preface and contents.
Recitations: Tuesday at the following times and locations.
- R01: 9am with M. Chen (66-144)
- R02: 10am with M. Chen (66-144)
- R03: 10am with V. Krylov (2-136)
- R04: 11am with D. Kliuev (4-159)
- R05: 11am with V. Krylov (2-136)
- R06: 12pm with M. Harris (4-159)
- R07: 12pm with D. Kliuev (2-105)
- R08: 12pm with I. Ganguly (2-136)
- R09: 1pm with M. Harris (4-149)
- R10: 1pm with I. Ganguly (2-136)
- R11: 1pm with Z. Chen (2-135)
- R12: 2pm with Z. Chen (2-132)
- R13: 2pm with K. Vashaw (2-136)
- R14: 3pm with K. Vashaw (2-136)
Office Hours: Please make arrangements to meet Professor Strang before or after class if you wish. The rest of the instructional team will hold weekly office hours. You may attend any office hours that fit your schedule.
- A. Horning: Friday 12:30 - 1:30 in 66-144
- M. Chen: Friday 4-6 in 66-154
- V. Krylov: Thursday 5 - 7 in 2-361
- D. Kliuev: Thursday 11 - 1 in 2-341
- M. Harris: Wednesday 4 - 6 in 2-333
- I. Ganguly: Monday 12:30-1:30 in 2-146 and Thursday 1-2 in 2-139
- Z. Chen: Wednesday 12 - 2 in 2-340
- K. Vashaw: Monday 1:30 - 2:30 in 2-131 and Friday 1:30 - 2:30 in 66-144
- S. Bentley: Friday 3 - 4 in 2-147
- A. Lu: Friday 7 - 9 in 2-139
Exams: We will have 3 exams during the semester: February 22, March 20, April 19. Final Exam on May 22 (official schedule here). For conflicts and/or accomodations, please contact Sapphire Tang in Academic Services as soon as possible (1-2 weeks in advance).
Homework: Due weekly on Sunday at midnight (except exam weeks and spring break). Upload a .pdf of your clearly written or typed solutions on Gradescope. Late homework will not be accepted and extensions will not be granted within 48 hours of the due date except in cases of genuine emergency (a letter from S^3 is required).
Collaboration: Collaboration on homework is highly encouraged! However, please maintain academic integrity by writing up your solutions individually and by naming all collaborators and information sources consulted for the assignment (you don't need to cite the textbook).
Grading: 20% Homeworks (equally weighted, lowest dropped) + 45% Quizzes (3 midterms, each worth 15%) + 35% Final Exam
Resources: In addition to this repository, we will use the following online resources.
- Canvas - course announcements will be posted on Canvas.
- Gradescope - submit Psets and check grades through Gradescope.
- Piazza - ask questions in the course discussion forum.
MIT also has excellent study resources: math learning center, TSR^2 study/resource room, pset partners.
-
Pset 1 is due on Sunday February 12 at 11:59pm.
-
Pset 2 is due on Sunday February 19 at 11:59pm.
-
Pset 3 is due on Sunday March 5 at 11:59pm.
-
Pset 4 is due on Sunday March 12 at 11:59pm. Extended to Wednesday March 15 at 11:59pm.
-
Pset 5 is due on Sunday April 9 at 11:59pm. Problem 9 is OPTIONAL for EXTRA CREDIT.
-
Pset 6 is due on Sunday April 16 at 11:59pm.
-
Pset 7 is due on Sunday April 30 at 11:59pm.
-
Pset 8 is due on Sunday May 7 at 11:59pm.
-
Exam 1 will be held on Wednesday, February 22 between 11am-12pm. Last names beginning with A-L will be in 50-340, while last names beginning with M-Z will be in 26-100.
-
Exam 2 will be held on Monday, March 20 between 11am-12pm. Last names beginning with A-L will be in 50-340, while last names beginning with M-Z will be in 26-100.
-
Exam 3 will be held on Wednesday, April 19 between 11am-12pm. Last names beginning with A-L will be in 50-340, while last names beginning with M-Z will be in 26-100.
Two ways to do matrix-vector multiplication are the "row way" and the "column way."
- "Row way" = dot products between rows of matrix and the vector.
- "Column way" = linear combination of columns in the matrix. The coefficients in the linear combination are the entries of the vector.
The column way leads to the column space of a matrix, the space of all possible linear combinations of the columns. Does the column space of a 3 x 3 matrix form a line, a plane, or fill the whole space? It depends on how many linearly independent columns there are. The geometric picture of a matrix's column space is the first key idea of linear algebra.
The dot product tells us how to do matrix-vector multiplication the "row way." The remarkable cosine formula for the dot product also shows how to use it to measure the length of vectors and the angles between them. In particular, perpendicular vectors have dot product = 0. Two useful inequalities follow quickly from the cosine formula: the Cauchy-Schwarz and triangle inequalities. Beyond matrix multiplication, the dot product reveals the geometry of vectors and their linear combinations.
Further Reading: Textbook, chapter 1.1 and 1.2.
The number of linearly indepdent columns in a matrix is the column rank. The number of linearly independent rows is the row rank. The remarkable fact is that these two numbers are the same! In the rank one case, this means that all the columns are multiples of a single vector. The rows are also multiples of a single vector! We can write the rank one matrix A as the product of a column vector and row vector: the row vector tells us what multiples we use to get the columns of A.
Beyond rank one, we can select linearly independent columns of A by moving from left to right. We ask, is the next column a linear combination of the previous columns? The linearly independent columns of A become the columns of a matrix C. The columns of a matrix R tell us how to combine the columns of C to get ALL columns of A: this is the factorization A = C R. R always contains the identity matrix (unless A is zero) and its rows are linearly independent. With A = C R, we are very close to understanding why the column and row rank of A have to be the same!
Further Reading: Textbook, chapter 1.3 and 1.4.
There are four useful ways to organize matrix-matrix multiplication:
- dot product between row i of A and column j of B gives entry (i,j) of AB,
- matrix A times column j of B gives column j of AB,
- row i of A times matrix B gives row i of AB,
- AB is a sum of rank 1 matrices: columns of A times rows of B.
The middle two are particularly useful for understanding the column or row space of the product AB.
With these conceptual ways to organize matrix-matrix multiplication, we see that A = CR means that the columns of A are combinations of columns of C and the rows of A are combinations of the rows of R. The factors C and R reveal the column space and the row space of the matrix A.
Further Reading: Textbook, review chapter 1. You may find this review sheet helpful.
At the heart of linear algebra is the equation Ax = b. We have n equations in n unknowns and A is an n by n matrix.
- If A has rank n, there is a unique solution.
- If A has rank less than n, then there is either (a) no solution or (b) infinitely many solutions!
In a simple 2 by 2 example, the second scenario happens when the equations define parallel lines that (a) never intersect or (b) are the same line.
To solve Ax = b, we combine equations to eliminate unknowns. This is elimination: elementary row operations on A and b convert A to upper triangular form. Once A is in triangular form, we can solve one variable at a time with backsubstitution.
Further Reading: Textbook, chapter 2.1.
Elimination converts A to an upper triangular matrix U. We can write the elimination steps as multiplication on the left of A by elimination matrices. Gathering the product of the elimination matrices on the left of A gives us EA = U. Inverting E, we arive at the famous factorization A = LU: A is the product of a lower (L) and an upper triangular matrix (U).
Elimination and backsubstitution implicitly invert A to solve Ax = b. In practice, we rarely compute the inverse of A directly (e.g., calculate its entries). However, it's a useful theoretical tool: studying the properties of inverses will allow us to connect the ideas of elimination to the ideas of column and row space from week one.
Further Reading: Textbook, chapter 2.2 and 2.3.
The inverse of a matrix gives us a direct way to think about Ax = b. An invertible matrix has a unique inverse, a unique solution to Ax = b for any b, and its columns are linearly independent. A triangular matrix is invertible if and only if its diagonal entries are nonzero. The inverse of a product of matrices is the product of the inverses in reverse order!
We rarely compute inverses explicitly on the computer. Instead, we solve Ax = b with elimination and backsubstitution. The inverse of the product of elimination matrices has a special structure. The diagonal entries are all one and the subdiagonal entries are the multipliers from elimination! This is the lower triangular matrix L in A = LU. If we encounter unwanted zeros on the diagonal during elimination, we can (when A is invertible) remedy the situation by interchanging rows to move the offending zero and replace it with a nonzero pivot. This leads us to elimination with row-interchanges: PA = LU.
Further Reading: Textbook, chapter 2.2, 2.3, and 2.4.
The algebra of row interchanges is captured in a very special family, or group, of matrices called permutatation matrices. The product, inverse, and transpose of a permutation matrix are all also permutation matrices! When permuation matrices multiply a matrix (or vector) from the left, they exchange rows. When they multiply from the right, the exchange columns.
The tranpose of a matrix switches the rows and the columns. The tranpose of a column vector is a row vector and vice versa. The transpose of a product is the product of the tranposes, in reverse order. The dot product is a special case of matrix multiplication: the tranpose of x multiplies y. Going back to the law of associativity, the dot product of A times x with y is the dot product of x with A transpose times y.
A very special class of matrices has A transpose equal to A. The rows and columns of A are the same! Professor Strang's favorite matrix is the second central difference matrix: symmetric, tridiagonal, and invertible. It is often used to approximate the second derivative of a function, sampled at equispaced points, with the method of finite-differences.
Further Reading: Textbook, chapter 2.4 and 2.5.
EXAM 1: NO LECTURE.
Finite difference matrices approximate derivatives of a function from its samples on a finite grid: these are the difference quotients of calculus in the language of linear algebra. The second central different matrix is special: it is symmetric, tridiagonal, and invertible (with the right boundary conditions). It's LU factorization reflects this symmetry: it is a product of forward and backward difference matrices.
We can use difference quotients and finite difference matrices to solve differential equations on the computer! Discretizing the heat equation with finite differences and a backward difference quotient in time (implicit time-stepping), we have to solve a linear system of equations to step forward in time: each new time step gives a new right-hand side that depends on the solution from the previous time step. The coefficient matrix stays the same, so it is best to factor once (A=LU) and only solve triangular systems after that!
Further Reading: Textbook, Chapter 2.5.
A VECTOR SPACE has 2 operations: sum x + y of "vectors" and multiplication cx by "scalars." Then 8 rules like c(x + y) = cx + cy must hold. A "SUBSPACE" is a vector space INSIDE another vector space, as in these examples of subspaces of R^n = n-dimensional space:
- all combinations of k given vectors
- all solutions to Ax = 0 for a given matrix A
Similarly, we have subspaces of C^n with complex numbers and matrix spaces like all m by n real matrices and function spaces like all continuous functions f(x) for 0 <= x <= 1. All diagonal matrices and all symmetric matrices would be subspaces of the vector space of n by n matrices.
Further Reading: Textbook, Chapter 3.1.
Finding a complete set of solutions to Ax = zero vector for a given m by n matrix A. This is the "nullspace of A " - a subspace of n-dimensional space. We need elimination to simplify Ax = 0 to a "reduced echelon form Rx = 0." Suppose A is m by n of rank r (r independent rows and columns). Use row operations on A to produce R0 (m by n) and then R (m by r, the same R as in A = CR of Chapter 1).
1 2 11 17 1 0 3 5 1 0 3 5
3 7 37 57 reduces to R0 = 0 1 4 6 and then to R = 0 1 4 6
4 9 48 74 0 0 0 0
Delete the zero row and now A = CR.
Elimination is complete and we easily solve Rx = 0 (same nullspace as A). There is a special solution for each column of R apart from the r=2 columns of I. Set x3 = 1 and x4 = 0 to find the special solution x = ( -3, -4, 1 , 0) to Ax = 0. Set x3 = 0 and x4 = 1 to find the special solution x = ( -5, -6, 0, 1) to Ax = 0. The combinations of those special solutions fill the nullspace of A. We will soon show that this nullspace is perpendicular to the row space because Ax = 0.
Further Reading: Textbook, Chapter 3.2.
We're now ready to describe the complete solution to A * x=b. If A is square and invertible, there is a unique solution. If A is not invertible and b is not in the column space of A, there is no solution. If A is not invertible, but b is in the column space of A, we have infinitely many solutions! How do we describe them? The nullspace of A plays the key role.
If xp solves A * xp = b and xn solves A * xn = 0 (xn is in the nullspace of A), then A * (xp + xn) = b so x = xp + xn also solves A * x = b! Every solution of A * x = b can be written in this form: a particular solution xp, which solves A * xp = b, plus any vector xn in the nullspace of A. When A is invertible, the nullspace is trivial and the solution is unique. Otherwise, we can write down an infinite set of solutions because the nullspace contains infinitely many vectors!
** Further Reading:** Textbook, Chapter 3.3.
The big picture of linear algebra is the four fundamental subspaces of an m x n matrix A with rank r: the column spaces of A and its transpose (row space), and the nullspaces of A and its transpose. The row space (dimension r) and the nullspace (dimension n-r) of A are orthogonal complements in R^n. Every vector in R^n can be written as the sum of two orthogonal vectors: one in the row space and one in the null space. There is a beautiful symmetry here because the same picture holds for the column space of A and the nullspace of A^T!
A basis for a vector space is a spanning set of linearly independent vectors: any vector in the space can be written as a unique linear combination of basis vectors. The dimension of a vector space is the number of vectors in (any and every) basis for the space. The factorization A = C*R gives us a basis for the column space of A (columns of C), row space of A (rows of R), and the nullspace of A (constructed by solving Rx = 0: easy since R is already the result of elimination). How can we find a basis for the nullspace of A^T?
Further Reading: Textbook, Chapter 3.4, 3.5, and 4.1.
An incidence matrix A describes the flow of information on a directed graph, e.g., the flow of electricity in an electical circuit. The four fundamental subspaces of A provide clear insight into the structure and behavior of the circuit. Kirchoff's laws of current, voltage, and Euler's formula relating the number of nodes, edges, small loops in the graph: these can all be seen in the column spaces and nullspaces of A and A^T.
So far, we have learned how to solve and analyze Ax = b when A is square and invertible or when b is in the column space of A. But what should we do when b is not in the column space of A? This situation is typical of regression problems in classical statistics and data-science, when each data-point leads to an equation and one hopes to find model parameters that fit the data well. The first thing to try is to make Ax as close to b as possible: this means choosing x so that Ax-b is orthogonal to Ax. To find x and Ax-b, we need to study orthogonal projections onto the column space of A.
Further Reading: Textbook, Chapter 3.5, 4.1, and 4.2.
To make Ax-b as small as possible, we choose x so that Ax - b is orthogonal to the column space of A. Then, Pb = Ax is the orthogonal projection of b onto the column space of A. The error in Ax = b is b - Ax = b - Pb = (I-P)b. The projections P and (I-P) are called orthogonal projectors: they are square, symmetric, and their ranks are equal to the subspaces they project onto. The projector P projects onto the column space of A and I-P projects onto the orthogonal complement, the nullspace of A^T.
To find P and I - P, we have to find x! The condition Ax - b orthogonal to columns of A leads to the normal equations: (A^TA)x=A^Tb. This is a square linear system, and it is invertible when the columns of A are linearly independent.
Further Reading: Textbook, Chapter 4.1 and 4.2.
If the columns of A are indepedent (full column rank), then the normal equations (A^TA)x = A^Tb have a solution x that makes the vector Ax-b as short as possible! The vector p = Ax is the projection of b onto the column space of A: p = A * (A^T * A)^(-1) A^T * b. The matrix multiplying b from the left is the orthogonal projection matrix P. Applying the projection twice is the same as applying the projection once: P^2 = P.
We can use the normal equations to solve regression problems in statistics, like finding a line that best fits the data. Each data point gives us an equation, a row of Ax = b. The slope and intercept of the line are the unknowns, entries of x. To choose the slope and intercept of the best fitting line, we minimize ||Ax-b|| - we solve the normal equations! The error in our fit is ||Ax-b||=||b - Ax|| = ||(I-P)b||: this is how much of b is orthogonal to the column space of A.
Further Reading: Textbook, Chapter 4.2 and 4.3.
Orthogonal matrices are a beautiful family of square matrices with orthonormal columns: the columns are orthogonal to each other and have unit length. Examples of orthgonal matrices come from permutations, rotations, reflections, and Hadamard matrices (entires are plus and minus one). Orthogonal matrices preserve the length of a vector when they mutliply it. When Q is an orthogonal matrix, ||Qx||=||x||! Orthogonal matrices and orthogonal bases are the key to make orthogonal projections and least-squares work reliably on the computer.
Further Reading: Textbook, Chapter 4.4. Check out the least-squares notebook for more applications of least-squares.
To find an orthonormal basis (orthogonal and normalized to unit length) for the column space of A, we can perform the Gram-Schmidt orthogonalization procedure. The first k basis vectors from Gram-Schmidt form an orthonormal basis for the first k linearly independent columns of A. If A has linearly independent columns, we get the factorization A = QR. The columns of Q are the orthonormal basis for the column space of A and the columns of R tell us how to recover the columns of A from Q!
Further Reading: Textbook, Chapter 4.4.
Gramd-Schmidt orthogonalization provdes the factorization A = QR. The columns of Q form an orthonormal basis for the column space of A. The columns of R link the columns of Q to the columns of A. The entries of R are the dot products we compute during Gram-Schmidt. Once we have computed an orthonormal basis for the column space of A, we can use the factors Q and R to solve the least-squares problem. The new (simpler) equation for the least-squares solution is the upper triangular system Rx = Q^Tb. High-quality numerical linear algebra software usually works with Rx = Q^Tb instead of the normal equations in order to reduce the accumulation of rounding errors made during computer arithmetic.
Further Reading: Textbook, Chapter 4.4.
It's determinant week! The determinant of a square matrix provides a useful test for invertibility: it is zero exactly when the matrix is not invertible. The determinant of the identity is one, exchanging rows (or columns) changes the sign of the determinant, and the determinant is linear in each separate row and column. This last statement means that scaling a single row (or column) scales the whole determinant. The familiar and remarkable formualas for the determinant follow from these three properties!
Although historically important and theoretically insightful, we rarely use the determinant (or explicit formulas) for computing inverses: it is almost always best to solve linear systems by elimination. When the determinant must be computed, it is usually done via elimination and A=LU. Then, det(A) = det(L) * det(U), and the determinants of the triangular matrices L and U are the products of their diagonal elements.
Further Reading: Textbook, Chapter 5.1.
The three definining properties of the determinant lead to the famous product rule, det(AB) = det(A) * det(B). They also lead to the cofactor expansions, which reduce the determinant calculation to a combination of "one size smaller" determinants: the cofactors. The cofactors of a matrix are the key to explicit formulas for its inverse! The inverse of a matrix A is the matrix of cofactors C transposed and divided by the determinant: A^(-1) = C^T / det(A).
The cofactor formula for the inverse is often useful theoretically, but it does not lead to efficient or stable methods for numerical inversion or the solution of linear systems. However, the cofactor expansion also reveals that the determinant of a triangular matrix is the product of its diagonal elements. This leads to an elegant and practical method for computing determinants using A = L * U because det(A) = det(L) * det(U) = (product of pivots). When computing determinants numerically is explicilty required (rarely) in an application, we return to elimination and A = LU.
Further Reading: Textbook, Chapter 5.1 and 5.2.
The determinant has an elegant connection to geometry that makes it indispensible in certain areas of geometry and calculus. If we use the columns of a square n-by-n matrix A to fill out the sides of a parallelpiped in n dimensions, the determinant of A is equal to the area of that parallelpiped.
The eigenvectors of a matrix identify extremely special directions. When a matrix multiplies its eigenvector, the direction doesn't change! The result is a scaled ("strectched" or "shrunk") version of the same eigenvector. The scaling factor is the eigenvalue: if the eigenvalue is less than 1 in magnitude, the eigenvector shrinks. If the eigenvalue is greater than 1 in magnitude, the eigenvector stretches. A zero eigenvalue means the matrix is singular, while an eigenvalue with magnitude one means the matrix doesn't stretch or shrink the eigenvector at all.
Further Reading: Textbook, Chapter 5.3 and 6.1.
To compute eigenvalues and eigenvectors, we first use the det(A - lambda * I)=0 to find the eigenvalues. Once we have the eigenvalues lambda that make A - lambda * I singular (zero determinant), we solve for vectors in the nullspace of A - lambda * I. These are the eigenvectors, the solutions of Ax = lambda x.
The eigenvalues of a triangular matrix are just the values on the diagonal. The eigenvectors of a rank one matrix uv^T are u (eigenvalue = v^Tu) and all vectors orthogonal to v (eigenvalues = 0). The eigenvalues of an orthogonal matrix always have |lambda|=1 because orthogonal matrices don't change the lengths of vectors! The eigenvalues of the familiar 2 x 2 "rotation-by-theta" matrix might be a surprise: they are the complex numbers exp(i theta). The eigenvalues of a 2 x 2 matrix can be expressed simply in terms of the trace and determinant of the matrix using the quadratic formula. In any dimension, the trace of a matrix = the sum of the eigenvalues and the determinant of a matrix = the product of the eigenvalues.
Further Reading: Textbook, Chapter 6.1 and 6.2.
When the eigenvectors of A form a basis (n linearly independent vectors for n x n A), A acts like a diagonal matrix in that basis! We can switch to the eigenvector basis to break difficult problems into simpler pieces. For example, coupled linear differential equations in multiple variables become a collection of uncoupled linear differential equations that can be solved with the tools of 1D calculus. That is the power of eigenvalues and eigenvectors at work! In matrix notation, diagonalization is expressed as A = X D X^(-1), where the columns of X are the eigenvectors of A and the diagonal matrix D has the eigenvalues of A on its diagonal. In the eigenvector basis A becomes X A X^(-1) = D - a diagonal matrix! We say that A is diagonalizable. Every matrix with distinct eigenvalues (no repeats) is diagonalizable.
Further Reading: Textbook, Chapter 6.2 and 6.3.
Matrices that are symmetric, S^T = S, are a very special type of diagonalizable matrix. They have a full basis of orthonormal eigenvectors! And their eigenvalues are always real (remember that many matrices can have complex eigenvalues). Life is good when we are dealing with symmetric matrices.
Further Reading: Textbook, Chapter 6.3.
Although this course focuses primarily on matrices and vectors whose entries are real numbers, complex matrices and vectors play an incredibly important role in applied mathematics. Most of the important facts about real linear algebra have a mirror image in complex linear algebra, as long as we replace the transpose operation for real vectors and matrices by the conjugate transpose operation. Then, real symmetric matrices become complex Hermitian matrices, real orthogonal matrices become complex unitary matrices, and so on.
Further Reading: Textbook, Chapter 6.4.
Eigenvalues and eigenvectors play a key role in the solution and analysis of linear systems of differential equations. The eigenvectors define important invariant directions in the phase space. Any initial condition aligned with an eigenvector stays in that direction for all time! The eigenvalues reveal the dynamics along these invariant directions: positive eigenvalues lead to solutions that grow expontially in time, while negative eigenvalues lead to solutions that decay exponentially in time. The general solutions are linear combinations of the solutions along the invariant directions.
Further Reading: Textbook, Chapter 6.5.
The singular value decomposition (SVD) factors every matrix (square or rectangular) into the product of three simpler matrices: A = USV^T. U and V have orthonormal columns called left and right singular vectors, respectively, while S is the diagonal matrix of singular values. Geometrically, the SVD that every matrix can be diagonalized by rotating (or reflecting) its input and output spaces, i.e., with orthogonal transformations. At the heart of the SVD, one finds the eigenvalues and eigenvectors of the square symmetric positive definite matrices AA^T and A^TA.
Further Reading: Textbook, Chapter 7.1.
The SVD is constructed from eigenvalues and eigenvectors of the square symmetric positive definite matrices A^TA and AA^T. Symmetry means these matrices have complete orthogonormal sets of eigenvectors, which can be collected into orthogonal matrices U and V. Postive definite means that the eigenvalues are real and postive - in fact, both matrices have the same nonzero eigenvalues! The key to the SVD is the link between these two orthonormal sets of eigenvectors: they lead to the decomposition AV = US and then A = USV^T. The columns of U and V are orthogonal bases for the column and row spaces of A, respectively, and the singular values S describes how A stretches and shrinks vectors along these coupled input-output directions.
Further Reading: Textbook, Chapter 7.1 and 7.2.
The singular values of A describe the coupling strength between special input directions (columns of V - right singular vectors) and special output directions (columns of U - left singular vectors). The singular values reveal the important directions in the row and column spaces of A. We can build a low-rank opproximation to A by dropping the small singular values and associated singular vectors from the SVD. Storing the largest few singular values and vectors can provide very good approximations to matrices with rapidly decaying singular values at a fraction of the cost of storing A.
As a first example, consider the task of compressing an array of pixels in a digital image without losing too much image quality. The SVD of the array can provide remarkable compression power! As a second example, consider the problem of identifying directions of large variance in student or consumer data. Which combinations of variables account for the largest differences among students? The SVD leads to principle componant analysis (PCA), which identifies these directions and uses them to decouple covariant parameters.
Further Reading: Textbook, Chapter 7.2 and 7.3. Experiment wit the SVD for image compression!
On the computer, eigenvalues (and singular values) are not computed from det(A - lambda I). Instead, they are computed with an iterative method called the QR algorithm. The idea of the QR algorithm is simple: compute the QR factorization A=QR, reverse the factors and compute AA = RQ, compute the new QR factorization of AA, reverse the factors again, ...., and repeat until convergence! This extroardinary strategy reveals the eigenvalues of A along the diagonal of the upper triangular factors. With the right "bells and whistles," the QR iterations can be computed efficiently and each eigenvalues is revealed rapidly within a few iterations.
Many problems in applied math boil down to finding the minimum value of a function that depends on many variables. Often, we don't know the function explicitly, but we can evaluate it and (perhaps approximate) its partial derivatives. How should we go about finding the minimum? One strategy is to simply walk downhill. The gradient (vector of partial derivatives) tells us which way to walk - the direction of steepest descent. Gradient descent (also called steepest descent) is the basic building block for many first-order optimization algorithms that are used to solve engineering design problems, train neural networks, and much more.
Further Reading: Textbook, Chapter 9.1.
Gradient descent is the prototypical "first-order" optimization algorithm: the algorithm tries to find local minima by evaluating the objective function and its derivative to "walk downhill." Pure gradient descent can get stuck in a criss-crossing pattern in narrow valley's, where the function is ill-conditioned, and converge slowly. Adding an inertial term that accounts for the previous search direction can help mitigate slow convergence due to ill-conditioning.
In deep neural networks, complex and high-dimensional functions are approximated by a layered network of "neurons:" Each neuron has an "activation function" that governs its response to inputs. The inputs and ouputs of neurons in each layer are connected by linear (or affine) transformations. The weights of a neural network govern the strength of connections between individual neurons. The weights are adjusted while "training" the network, i.e., minimizing a loss function that describes how closely the neural network's predictions match the training data.
Further Reading: Textbook, Chapter 9.2
Deep neural networks are constructed by composing relatively simple parts (linear transformations and simple nonlinear activation functions) and composing them in complex architectures. Althuough classical statistics warns against overparametrizing models and "overfitting" noisy data, neural nets have led to the discovery that nonlinear models can actually thrive in the overparametrized regime. Understanding the approximation power of deep neural networks and their ability to generalize from training data to test data is a fascinating area of active research.
Further Reading: Explore the basics with this playground developed by Daniel Smilkov and Shan Carter.