1. Last lecture summary

2. Orthogonal matrices

3. Put orthonormal vectors into a matrix
independent basis, orthogonal basis, orthonormal vectors, normalization
Put orthonormal vectors into a matrix
Generally rectangular matrix – matrix with orhonormal columns
Square matrix with orthonormal colums – orthogonal matrix Q
Matrix A has non-orthogonal columns,
A → Q … Gram-Schmidt Orthogonalization
Q-1 = QT

4. Having Q simplifies many formulas. e. g.
QR decomposition: A = QR

5. Determinants

6. determinant is a test for invertibility
Properties:
Exchange rows/cols of a matrix: reverse sign of det
Equals to 0 if two rows/cols are same.
Det of triangular matrix is a product of diagonal terms.
determinant is a test for invertibility
matrix is invertible if |A| ≠ 0
matrix is singular if |A| = 0

7. Eigenvalues and eigenvectors

8. Action of matrix A on vector x returns the vector parallel (same direction) to vector x.
x … eigenvectors, λ … eigenvalues
spectrum, trace (sum of λ), determinant (product of λ)
λ = 0 … singular matrix
Find eigendecomposition by solving characteristic equation (leading to characteristic polynomial)
det(A - λI) = 0
Find λ first, then find eigenvectors by Gauss elimination, as the solution of (A - λI)x = 0.

9. algebraic multiplicity of an eigenvalue
multiplicity of the corresponding root
geometric multiplicity of an eigenvalue
number of linearly independent eigenvectors with that eigenvalue
degenerate, deffective matrix

10. Diagonalization
Square matrix A … diagonalizable if there exists an invertible matrix P such that P −1AP is a diagonal matrix.
Diagonalization … process of finding a corresponding diagonal matrix.
Diagonalization of square matrix with independent eigenvectors meas its eigendecomposition
S-1AS = Λ

11. Which matrices are diagonalizable (i.e. eigendecomposed)?
all λ’s are different (λ can be zero, the matrix is still diagonalizable. But it is singular.)
if some λ’s are same, I must be careful, I must check the eigenvectors.
If I have repeated eigenvalues, I may or may not have n independent eigenvectors.
If the eigenvectors are independent, than it is diagonalizable (even if I have some repeating eigenvalues).
Diagonalizability is concerned with eigenvectors. (independent)
Invertibility is concerned with eigenvalues. (λ=0)

12. symmetric matrix (n x n)
diagonal matrix
eigenvalues are on the diagonal
eigenvectors are columns with 0 and 1
triangular matrix
eigenvalues are sitting along the diagonal
symmetric matrix (n x n)
real eigenvalues, real eigenvectors
n independent eigenvectors
they can be chosen such that they are orthonormal (A = QΛQT)
orthogonal matrix
eigenvalues are +1 or -1

13. Symmetric matrices
eigenvalues are real, eigenvectors are orthogonal (may be chosen to be orthogonal)
Symmetric matrices with all the eigenvalues positive are called positive definite (semidefinite).
A linear system of equations with a positive definite matrix can be efficiently solved using the so-called Cholesky decomposition.
M = LLT = RTR (L – lower triangular, R – upper)
ATA is always symmetric positive definite provided that all columns of A are independent.

14. Singular Value Decomposition
based on excelent video lectures by Gilbert Strang, MIT
Lecture 29

15. SVD demonstration
we have vectors x and y perpendicular, find such Ax and Ay that are also perpendicular
If found, let’s label x as v1 and y as v2. And Ax as u1 and Ay as u2.
If v1 and v2 are orthonormal, u1 and u1 are orthogonal.
But can be normalized – multiplied by numbers σ1 and σ2

16. So matrix A acts on vector v1 and vector σ1u1 is obtained → Av1 = σ1u1
And similarly Av2 = σ2u2
Generally, this can be written in a matrix form as
AV = UΣ
But realize, that columns of V and U are orthonormal. U and V are orthogonal matrices!
AV = UΣ → A = UΣV-1 → A = UΣVT

17. Numbers σ are called singular values.
And A = UΣVT is called singular value decomposition (SVD).
Factors: orthogonal matrix, diagonal matrix, orthogonal matrix – all good matrices!
Σ – singular values (diagonal)
columns of U – left singular vectors (orthogonal)
columns of V – right singular vectors (orthogonal)
Any matrix has SVD!
that’s why we actually need two orthogonal matrices
columns of V means rows of VT

18. Dimensions in SVD
A: m x n
U: m x m
V: n x n
Σ: m x n

19. Consider case of square full-rank matrix Am x m.
Look again at Av1 = σ1u1
σ1u1 is a linear combination of columns of A. Where lies σ1u1?
In C(A)
And where lies v1?
In C(AT)
In SVD I’m looking for an orthogonal basis in row space of A that gets knocked over into an orthogonal basis in column space of A.
- A: m x n
- How many components has v1? n components

20. Can I construct an orthogonal basis of C(AT)?
Of course I can, Gram-Schmidt does this.
However, when I take such a basis and multiply it by A, there is no reason why the result should also be orthogonal.
I am looking for the special setup, where A turns orthogonal basis vectors from row space into orthogonal basis in the column space.

21. In case A is symmetric positive definite, the SVD becomes A = QΛQT.
basis vectors
in row space
of A
basis vectors
in column space
of A
multiplying
factors
In case A is symmetric positive definite, the SVD becomes A = QΛQT.
SVD of positive definite matrix directly reveals its eigendecomposition.

22. SVD of 2 x 2 nonsingular matrix
U, Σ, V are all of 2 x 2
If A is real, U and V are also real.
The singular values σ are always real, nonegative, and they are conventionally arranged in descending order on the main diagonal of Σ.

23. SVD of 2 x 2 singular matrix
SVD is rank revealing factorization. rank(A) is the number of nonzero singular values.
Due to the experimental errors matrices are close to singular – rank deficient
SVD helps in this situation by revaling the intrinsic dimension. Close to zero singular values can then be set equal to zero.
- rank deficient – also rank deffective

24. In this example 4 x 4 matrix is generated, with 4th column almost equal to c1 – c2. Then the 4th singular is zeroed, and matrix is reconstructed.
Matlab code
c1=[ ]'
c2=[ ]
c3=[ ]'
c4 = c1-c *(rand(4,1))
A = [c1 c2 c3 c4]
format long
[U,S,V]=svd(A)
U1 = U; S1=S; V1 = V;
S1(4,4)=0.0
B=U1*S1*V1'
A-B
norm(A-B)
help norm
Extension of vector norms to matrices. It is a nonegative number ||A|| with properties: ||A||>=0, ||kA||=|k| ||A|| for scalar k, ||A+B|| <= ||A|| + ||B||
The analysis of matrix-based algorithms often requires use of matrix forms. These algorithms need a way to quantify the “size” of a matrix or the “distance” between two matrices.

25. m x n, m > n (overedetermined system)
with all columns independent
A = [c1 c2 c3]
[U,S,V]=svd(A)
U(1:3,1:3)*S(1:3,1:3)*V'
Of course, if we had also some columns dependent, then last singular values on the diagonal would be zero.
complete this to an orthonormal basis for the whole column space Rm
i.e. these vectors come from
left null space, which is
orthogonal to column space
complete this by
adding zeros

26. For the system m ≥ n, there exist a version of SVD, called thin or economy SVD. The zero block from Σ is removed, as well as are corresponding columns from U.
Dimensions of thin SVD
A: m x n
U: m x n
V: n x n
Σ: n x n
these are removed

27. m x n, n > m (underdetermined system)
with all rows independent
A = [c1(1:3) c2(1:3) c3(1:3) [ ]']
[U,S,V]=svd(A)
U*S(1:3,1:3)*V(1:3,1:3)'
complete this to an orthonormal basis for the whole row space Rn
i.e. these vectors come from
null space, which is
orthogonal to row space
complete this by
adding zeros

28. SVD chooses the right basis for the 4 subspaces AV=UΣ
Basis of C(AT)
Basis of C(A)
Basis of N(A)
Basis of N(AT)
SVD chooses the right basis for the 4 subspaces
AV=UΣ
v1…vr: orthonormal basis in Rn for C(AT)
vr+1…vn: in Rn for N(A)
u1…ur: in Rm for C(A)
ur+1…um: in Rm for N(AT)

29. Uniqueness of SVD non-equal singular values (non-degenerate)
U, V are unique except for the signs in columns
Once you decide signs for U (V), signs for V (U) are given so that A = UΣVT is guaranteed
equal non-zero singular values (degenerate)
not unique
if u1 and u2 correspond to the degenerate σ, also their any normalized linear combination is a left singular vector corresponding to σ
the same is true for right singular vectors

30. Uniqueness of SVD zero singular values
Corresponding columns of U and V are added
They form basis of N(AT) or N(A).
They are orthonormal each to other and to the rest of right/left singular vectors.
And are not unique.

31. How do we find SVD? A = UΣVT
I’ve got two orthogonal matrices and I don't want to find them both at once.
I need some expression so U disappears.
Let’s do ATA (nice positive (semi)definite symmetric). What’s the result?
ATA = VΣTUTUΣVT = VΣ2VT
This is actually a diagonalization for
positive definite ATA = QΛQT
V is found by diagonalizing ATA

32. Look at diagonalization of AAT, so V disappears.
How to find U?
Look at diagonalization of AAT, so V disappears.
Or compute 𝑢 1 = 1 𝜎 1 𝐴 𝑣 1
So u’s are eigenvectors of ATA, v’s are eigenvectors of AAT.
And eigenvalues are square roots of σ’s.
They are same for ATA and AAT.
They are unique.
They have arbitrary order. However, they are conventionally sorted in decreasing order (u, v must be reordered accordingly)

33. Example
Rank is?
two, invertible, nonsingular
I'm going to look for two vectors v1 and v2 in the row space, which of course is what? R2
And I'm going to look for u1, u2 in the column space, which is also R2, and I'm going to look for numbers σ1, σ2 so that it all comes out right.

34. SVD example 1st step – ATA
2nd step – find its eigenvectors (they’ll b the vs) and eigenvalues (squares of the σ)
eigen can be guessed just by looking
what are eigenvectors and their corresponding eigenvalues?
[1 1]T, λ1 = 32
[-1 1]T, λ2 = 18
Actually, I made one mistake, do you see which one?
I didn’t normalize eigenvectors, they should be [1/sqrt(2), 1/sqrt(2)]T …

35. 3rd step – AAT, find the u What are the eigenvectors? [1 0]T, [0 1]T
It’s diagonal, but just
by an accident

36. SVD expansion
Do you remember column times row picture of matrix multiplication?
If you look at the UΣVT multiplication as at the column times row problem, then you get the SVD expansion.
This is a sum of rank-one matrices. Each of uiviT is called mode.

37. This expansion is used in data compression application of SVD
This expansion is used in data compression application of SVD. We investigate the spectrum of singular values, and based on it we can decide when to stop the expansion.
This actually means that from Σ we remove singular values from the end, and that we remove coresponding columns from U and V.

38. SVD image compression load clown colormap('gray') image(X)
[U,S,V] = svd(X);
plot(diag(S));
size(X,1)*size(X,2)
p=1;image(U(:,1:p)*S(1:p,1:p)*V(:,1:p)');
size(U,1)*p+size(V',1)*p
To reconstruct Ap, we need to store only (m+n)·p words, compared
to m·n needed for storage of the full matrix A.
Please note, that if you put p = m (for m<n), then we need m2+m·n,
which is much more than m·n. So there is apparently some border
at which storing SVD matrices increases the storage needs!
example from Demmel

39. Numerical Linear Algebra Matrix Decompositions
based on excelent video lectures by Gilbert Strang, MIT
Lecture 29

40. Conditioning and stability
two fundamental issues in numerical analysis
Tell us something about the behavior after introduction of small error.
Conditioning – perturbation behavior of a mathematical problem.
Stability - perturbation behavior of an algorithm.

41. Conditioning
System’s behavior, it has nothing to do with numerical errors (round off).
Example: Ax = b
well-conditioned - small change in A or b results in a small change in the solution b
ill-conditioned - small change in A or b results in a large change in the solution b
Conditioning is quantified by the condition number, by its coupling with machine epsilon (precission) we can then quantify how many significant digits we can trust in the solution.
- from

42. well-conditioned system
small change in result b
ill-conditioned system
small change in coefficient matrix A
- from

43. Stability Algorithm’s behavior, involves rounding errors etc. Example:
our computer only has two digits of precision (i.e. only two nonzero numbers are allowed)
we have array of 100 numbers [1.00, 0.01, 0.01, 0.01, …]
and we want to sum them
mathematically exact solution is 1.99

44. So we take the first number (1. 00), then add another (1
So we take the first number (1.00), then add another (1.01), then another …. (1.10), then another (1.11, however, this can’t be stored in two-dogit precission, so computer rounds down to 1.10), …, so the final result is 1.10 (compare to correct result of 1.99)
Or we can sort the array first in ascending order [ , …, 1.00] and then sum: 0.01, 0.02, 0.03, …0.99, 1.99, 1.99 gets round to 2.00 (compare to correct result of 1.99)
unstable
stable

45. Matrix factorizations/decompositions
factorization - decomposition of an object into a product of other objects - factors - which when multiplied together give the original .
matrix factorization – product of some nicer/simpler matrices that retain particular properties of the original matrix A
nicer matrices: orthogonal, diagonal, symmetric, triangular
- see also

46. System of equations Ax = b
System of linear equations
It is solved by Gaussian elimination
Gaussian elimination leads to the LU factorization
U is upper traiangular, L is lower triangular with units on the diagonal

47. Ax = b … LUx = b, we can say Ux = y, and solve Ly = b and then Ux = y.
Why is it advantageous?
Similarly, Ux = y is solved by backward substitution.
LU factorization is Gauss elimination, so when should I do LU factorization?
If I have more bs, but A does not change. I do LU just once, and then I repeatedly change just RHS (b).
forward substitution

48. LU factorization is used to calculate matrix determinant.
A = LU, det(A) = det(LU) = det(L)det(U) = product of diagonal elements L (one) x product of diagonal elements U
Use of QR factorization for Ax = b
A = QR, QRx = b → Rx = QTb
Compute A = QR
Compute y = QTb
solve Rx = y by back-substitution

49. However the standard way for Ax = b is Gauss (LU), it requires half of the operations than QR.
LU factorization can also be used for matrix inversion (however, standard way is Gauss-Jacobi), L and U are easier to invert.

50. Least squares algorithms
ATAx = ATb
Least squares via normal equations:
Cholesky factorization ATA = RTR leads to RTRx = ATb
form ATA and ATb
compute Cholesky ATA = RTR
solve lower triangular system RTw = ATb for w
solve upper triangular system Rx = w for x

51. Least squares with QR factorization
ATAx = ATb
QR factorization of A leads to
(QR)TQRx = (QR)Tb → Rx = QTb
compute A = QR
compute the vector QTb
solve upper triangular system Rx = QTb for x
This is method of choice for least squares.

52. Eigenvalue algorithms
Eigenvalue decomposition A = SΛS-1
alternatively AS = SΛ
this leads to characteristic equation - polynomial (det λI – A = 0)
However, finding roots of polynomial is instable, it can be done for 2 x 2 or 3 x 3, but not for 50 x 50
Thus, this decomposition is usually not adopted for eigenproblems.

53. Eigenproblem via iterative QR algorithm Will be shown without proof:
Other iterative algorithms exist for special purposes (get first 30 eigenvalues, get eigenvalues from the given interval, non-symmetric matrices)
A(0) = A
for k = 1, 2, …
Q(k)R(k) = A(k-1) QR factorization of A(k-1)
A(k) = R(k)Q(k) Recombine factors in reverse order
- not QR decomposition (though it forms the basis of the algorithm)