# Matrix factorization

1. Gauss-Jordan elimination, for calculating inverse, leads to $A = LU$
2. Gram-Schmidt transformation, for finding orthonormal bases, leads to $A = QR$
3. Via eigenvalues and eigenvectors: $A = S \Lambda S^{-1}$, where $S$ is the matrix of eigenvectors and $\Lambda$ is a diagonal matrix of eigenvalues

# Three ways for calculating determinants

1. Multiply the $n$ pivots, the pivot formula. (Pivots are just from elimination, no scaling)
2. Add up $n!$ terms, the big formula. The sum of $n!$ simple determinants: the product of items in a matrix which is formed by picking an iterm from each row and column of the original matrix, and times $1$ or $-1$ depending on the number of permutations contained in the column numbers following the row order.
3. Combine $n$ smaller determinants, the cofactor formula: $\mathrm{det} A = a_{11}C_{11} + a_{12}C_{12} + \cdots + a_{1n}C_{1n}$. Note that only one row is involved! More generally, $\mathrm{det} A = a_{i1}C_{i1} + a_{i2}C_{i2} + \cdots + a_{in}C_{in}$, where $C_{ij} = (-1)^{i + j} \mathrm{det} M_{ij}$, so the cofactor has to include its correct sign!

# Properties of determinants

1. The determinant of the $n \times n$ matrix is 1
2. The determinant changes sign when two rows are exchanged
3. The determinant is a linear function of each row separately
4. If two rows of $A$ are equal, the $\mathrm{det} A = 0$
5. Subtracting a multiple of one row from another row leaves determinant unchanged
6. A matrix with a row of zeros has $\mathrm{det} A = 0$
7. If $A$ is triangular, then $\mathrm{det} A = a_{11} a_{22} \cdots a_{nn}$, the product of diagonal entries
8. If $A$ is singular, then $\mathrm{det} A =0$. If A is invertible, then $\mathrm{A} \neq 0$
9. The determinant of $AB$ is the $\mathrm{det} A$ times $\mathrm{det} B$: $\vert AB \vert = \vert A \vert \vert B \vert$
10. Transpose does not change determinant: $\vert A \vert = \vert A^T \vert$

When the first three are satisfied, the others follow.

# Eigenvalue and eigenvectors

1. A matrix cannot be diagonalized without $n$ independent eigenvectors
2. Real matrixes can easily have complex eigenvalues and eigenvectors

# Linking pivots, determinants, eigenvalues, and least squares

When a symmetric matrix has any of the following five properties , it has all of them:

1. All $n$ pivots are positive
2. All $n$ upper left determinants are positive
3. All $n$ eigenvalues are positive
4. All $x^TAx$ is positive except at $x=0$. This is the energy-based definition
5. $A$ equals $R^T R$ for a matrix $R$ with independent columns

Reference: Introduction to Linear Algebra (4th edition, 2009) by Gilbert Strang.