Eilf: Matrix stuff
Eilf: Matrix stuff
- Determinant of a matrix
- Difference between inverse matrix and identity matrix and what are they?
- Eigenvalues
- Unitary or orthonormal matrix
- Diagonal matrix
- How to compute matrices?
Thank you in advance for answering anyone of them.
- A good way to think of matrices is as a kind of function. They take column vectors as "input” by multiplying with them, and the "output" from that product is another vector. The determinant measures how much a matrix stretches the space the input vectors come from. Big determinants stretch spaces way out, small ones shrink them way down, and negative ones reverse them like a mirror.
@meowmeowmeow
2(a). In a lot of mathematical systems, the "identity" is the thing that "does nothing." For example, when adding ordinary numbers the identity is 0 because adding 0 to any number does nothing - the other number stays the same. Similarly, when multiplying the identity is 1 because multiplying 1 with any number also does nothing. The identity matrix plays the same role - if you multiply any (square) matrix with the identity, you'll get back the same matrix you started with.@meowmeowmeow
2(b). The inverse is related to the identity. It's sort of the "opposite" of a math object (a number, matrix, etc.) but in a specific way. When combining something with its inverse by some operation (like adding or multiplying) the result is the identity. For example: when adding, the inverse of x is -x because x+(-x) = 0. And when multiplying, the inverse of x is 1/x because x*1/x = 1. In the same way, when a matrix multiplies with its inverse, the result is the identity matrix.