Lemdro.id

4 points

1 year ago

@meowmeowmeow

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.

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4 points

1 year ago

@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.

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4 points

1 year ago

@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.

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2 points

1 year ago

@meowmeowmeow
3. Remember a matrix is like a function: multiply it with a column vector as input, and you get another column vector as output. In general, a matrix can transform vectors in all sorts of ways, but sometimes a matrix has special input vectors called “eigenvectors.” What makes them special is that, after multiplying, you get almost exactly the same vector you started with, but multiplied by some number called an “eigenvalue.” This page has some examples: https://www.mathsisfun.com/algebra/eigenvalue.html

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3 points

1 year ago

@meowmeowmeow
4(a). “Orthonormal” combines “orthogonal” (sort of means the same as “perpendicular”) and “normal” (in this context means a vector with length 1). If a matrix is orthonormal, that means if we treat its columns as separate vectors, they’re all mutually perpendicular to each other and each have length 1. Why do we care enough to give this a special name? Well, it turns out orthonormal matrices rotate and reflect vectors, which has obvious uses to science and computer graphics.

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3 points

1 year ago

@meowmeowmeow
4(b). An equivalent property of an orthonormal matrix is that its transpose (flipping a matrix so that every row becomes a column and every column a row) is equal to its inverse. Unitary matrices are almost exactly the same, except that they use complex numbers instead of just real ones, and instead of taking the transpose to get the inverse you also have to take the complex conjugate of every element. There’s a lot more to them, but this is the best way I can keep it ELI5.

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3 points

1 year ago

@meowmeowmeow
5. A diagonal matrix is what it sounds like - all of the (nonzero) entries are on the diagonal, from the top left corner to the bottom right. Why do we care? All sorts of calculations are easier with diagonal matrices, which is great for lazy mathematicians and efficient programmers. Some matrices aren’t diagonal, but “diagonalizable,” meaning we can shuffle them around into a similar diagonal matrix by using their eigenvectors, which comes in quite handy.

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meowmeowmeow@lemmy.mlOP

1 point

1 year ago

Thanks for your explaination with examples. What is a column vector? Is it something like (1 2 3) which means move x upwards 1 unit, y up 2 units, z up 3 units?

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3 points

1 year ago

@meowmeowmeow
Ah, I should have been more specific, but you pretty much have the right idea. A vector is, abstractly, something with a length and a direction, like a velocity or force in physics. But to actually make calculations with vectors it helps to represent them with lists of numbers like your example. The convention is that we write vectors vertically, hence “column vector.” Writing them horizontally as rows instead represents “covectors,” but I won’t get into the weeds on that.

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