In my linear algebra class, we just talked about determinants. So far I’ve been understanding the material okay, but now I’m very confused. I get that when the determinant is zero, the matrix doesn’t have an inverse. I can find the determinant of a $2\times 2$ matrix by the formula. Our teacher showed us how to compute the determinant of an $N \times N$ matrix by breaking it up into the determinants of smaller matrices, and apparently there is a way by summing over a bunch of permutations. But the notation is really hard for me and I don’t really know what’s going on with them anymore. Can someone help me figure out what a determinant is, intuitively, and how all those definitions of it are related?
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Your trouble with determinants is pretty common. They’re a hard thing to teach well, too, for two main reasons that I can see: the formulas you learn for computing them are messy and complicated, and there’s no “natural” way to interpret the value of the determinant, the way it’s easy to interpret the derivatives you do in calculus at first as the slope of the tangent line. It’s hard to believe things like the invertibility condition you’ve stated when it’s not even clear what the numbers mean and where they come from. Rather than show that the many usual definitions are all the same by comparing them to each other, I’m going to state some general properties of the determinant that I claim are enough to specify uniquely what number you should get when you put in a given matrix. Then it’s not too bad to check that all of the definitions for determinant that you’ve seen satisfy those properties I’ll state. The first thing to think about if you want an “abstract” definition of the determinant to unify all those others is that it’s not an array of numbers with bars on the side. What we’re really looking for is a function that takes N vectors (the N columns of the matrix) and returns a number. Let’s assume we’re working with real numbers for now. Remember how those operations you mentioned change the value of the determinant? (1) Switching two rows or columns changes the sign. (2) Multiplying one row by a constant multiplies the whole determinant by that constant. (3) The general fact that number two draws from: the determinant is linear in each row. That is, if you think of it as a function $\det: \mathbb{R}^{n^2} \rightarrow \mathbb{R}$, then $ \det(a \vec{v_1} +b \vec{w_1}, \vec{v_2},...,\vec{v_n}) = a \det(\vec{v_1},\vec{v_2},...,\vec{v_n}) + b \det(\vec{w_1}, \vec{v_2}, ...,\vec{v_n})$, and the corresponding condition in each other slot. I claim that these facts, together with the fact that the determinant of the identity matrix is one, is enough to define a unique function that takes in N vectors (each of length N) and returns a real number, the determinant of the matrix given by those vectors. I won’t prove that, but I’ll show you how it helps with some other interpretations of the determinant. In particular, there’s a nice geometric way to think of a determinant. Consider the unit cube in N dimensional space: the set of vectors of length N with coordinates 0 or 1 in each spot. The determinant of the linear transformation (matrix) T is the signed volume of the region gotten by applying T to the unit cube. (Don’t worry too much if you don’t know what the “signed” part means, for now). How does that follow from our abstract definition? Well, if you apply the identity to the unit cube, you get back the unit cube. And the volume of the unit cube is 1. If you stretch the cube by a constant factor in one direction only, the new volume is that constant. And if you stack two blocks together aligned on the same direction, their combined volume is the sum of their volumes: this all shows that the signed volume we have is linear in each coordinate when considered as a function of the input vectors. Finally, when you switch two of the vectors that define the unit cube, you flip the orientation. (Again, this is something to come back to later if you don’t know what that means). So there are ways to think about the determinant that aren’t symbol-pushing. If you’ve studied multivariable calculus, you could think about, with this geometric definition of determinant, why determinants (the Jacobian) pop up when we change coordinates doing integration. Hint: a derivative is a linear approximations of the associated function, and consider a “differential volume element” in your starting coordinate system. It’s not too much work to check that the area of the parallelogram formed by vectors $(a,b)$ and $(c,d)$ is $\det((a,b),(c,d))$, either: you might try that to get a sense for things. |
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You could think of a determinant as a volume. Think of the columns of the matrix as vectors at the origin forming the edges of a skewed box. The determinant gives the volume of that box. For example, in 2 dimensions, the columns of the matrix are the edges of a rhombus. You can derive the algebraic properties from this geometrical interpretation. For example, if two of the columns are linearly dependent, your box is missing a dimension and so it's been flattened to have zero volume. |
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In addition to the answers, above, the determinant is a function from the set of set of square matrices into the real numbers that preserves the operation of multiplication: \begin{equation}\det(AB) = \det(A)\det(B) \end{equation} and so it carries $some$ information about square matrices into the much more familiar set of real numbers. Some examples: The determinant function maps the identity matrix $I$ to the identity element of the real numbers ($\det(I) = 1$.) Which real number does not have a multiplicative inverse? The number 0. So which square matrices do not have multiplicative inverses? Those which are mapped to 0 by the determinant function. What is the determinant of the inverse of a matrix? The inverse of the determinant, of course. (Etc.) This "operation preserving" property of the determinant explains some of the value of the determinant function and provides a certain level of "intuition" for me in working with matrices. |
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The top exterior power of an $n$-dimensional vector space $V$ is one-dimensional. Its elements are sometimes called pseudoscalars, and they represent oriented $n$-dimensional volume elements. A linear operator $f$ on $V$ can be extended to a linear map on the exterior algebra according to the rules $f(\alpha) = \alpha$ for $\alpha$ a scalar and $f(A \wedge B) = f(A) \wedge f(B), f(A + B) = f(A) + f(B)$ for $A$ and $B$ blades of arbitrary grade. Trivia: some authors call this extension an outermorphism. The extended map will be grade-preserving; that is, if $A$ is a homogeneous element of the exterior algebra of grade $m$, then $f(A)$ will also have grade $m$. (This can be verified from the properties of the extended map I just listed.) All this implies that a linear map on the exterior algebra of $V$ once restricted to the top exterior power reduces to multiplication by a constant: the determinant of the original linear transformation. Since pseudoscalars represent oriented volume elements, this means that the determinant is precisely the factor by which the map scales oriented volumes. |
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For the record I'll try to give a reply to this old question, since I think some elements can be added to what has been already said. Even though they are basically just (complicated) expressions, determinants can be mysterious when first encountered. Questions that arise naturally are: (1) how are they defined in general?, (2) what are their important properties?, (3) why do they exist?, (4) why should we care?, and (5) why does their expression get so huge for large matrices? Since $2\times2$ and $3\times3$ determinants are easily defined explicitly, question (1) can wait. While (2) has many answers, the most important ones are, to me: determinants detect (by becoming 0) the linear dependence of $n$ vectors in dimension $n$, and they are an expression in the coordinates of those vectors (rather than for instance an algorithm). If you have a family of vectors that depend (or at least one of them depends) on a parameter, and you're asking (or are being asked) for which parameter values they are linearly dependent, than trying to use Gaussian elimination or something similar to detect linear dependence can run into trouble: one might need assumptions on the parameter to assure some coefficient is nonzero, and even then dividing by it gives very messy expressions. Provided the number of vectors equals the dimension $n$ of the space, taking a determinant will however immediately transform the question into an equation for the parameter (which one may or may not be capable of solving, but that is another matter). This is exactly how one obtains an equation in eigenvalue problems, in case you've seen those. This provides a first answer to (4). (But there is a lot more you can do with determinants once you get used to them.) As for question (3), the mystery of why determinants exist in the first place can be reduced by considering the situation where one has $n-1$ given linearly independent vectors, and asks when a final unknown vector $\vec x$ will remain independent from them, in terms of its coordinates. The answer is that it usually will, in fact always unless $\vec x$ happens to be in the linear span $S$ of those $n-1$ vectors, which is a subspace of dimension $n-1$. For instance, if $n=2$ (with one vector $\vec v$ given) the answer is "unless $\vec x$ is a scalar multiple of $\vec v$". Now if one imagines a fixed (nonzero) linear combination of the coordinates of $\vec x$ (the technical term is a linear form on the space), then it will become 0 precisely when $\vec x$ is in some subspace of dimension $n-1$. With some luck, this can be arranged to be precisely the linear span $S$. (In fact no luck is involved: if one extends the $n-1$ vectors by one more vector to a basis, then expressing $\vec x$ in that basis and taking its final coordinate will define such a linear form; however you can ignore this argument unless you are particularly suspicious.) Now the crucial observation is that not only does such a linear combination exist, its coefficients can be taken to be expressions in the coordinates of our $n-1$ vectors. For instance in the case $n=2$ if one puts $\vec v={a\choose b}$ and $\vec x={x_1\choose x_2}$, then the linear combination $-bx_1+ax_2$ does the job (it becomes 0 precisely when $\vec x$ is a scalar multiple of $\vec v$), and $-b$ and $a$ are clearly expressions in the coordinates of $\vec v$. In fact they are linear expressions. For $n=3$ with two given vectors, the expressions for the coefficients of the linear combination are more complicated, but they can still be explicitly written down (each coefficient is the difference of two products of coordinates, one form each vector). These expressions are linear in each of the vectors, if the other one is fixed. Thus one arrives at the notion of a multilinear expression (or form). The determinant is in fact a multilinear form: an expression that depends on $n$ vectors, and is linear in each of them taken individually (fixing the other vectors to arbitrary values). This means it is a sum of terms, each of which is the product of a coefficient, and of one coordinate each of all the $n$ vectors. But even ignoring the coefficients, there are many such terms possible: a whopping $n^n$ of them! However, we want an expression that becomes 0 when the vectors are linearly dependent. Now the magic (sort of) is that even the seemingly much weaker requirement that the expression becomes 0 when two successive vectors among the $n$ are equal will assure this, and it will moreover almost force the form of our expression upon us. Multilinear forms that satisfy this requirement are called alternating. I'll skip the (easy) arguments, but an alternating form cannot involve terms that take the same coordinate of any two different vectors, and they must change sign whenever one interchanges the role of two vectors (in particular they cannot be symmetric with respect to the vectors, even though the notion of linear dependence is symmetric; note that already $-bx_1+ax_2$ is not symmetric in $(a,b)$ and $(x_1,x_2)$). Thus any one term must involve each of the $n$ coordinates once, but not necessarily in order: it applies a permutation of the coordinates 1,2,...,$n$ to the successive vectors. Moreover, if a term involves one such permutation, then any term obtained by interchanging two positions in the permutation must also occur, with an opposite coefficient. But any two permutations can be transformed into one another by repeating such interchanges, so if there are any terms at all, then there must be terms for all $n!$ permutations and their coefficients are all equal or opposite. This explains question (5), why the determinant is such a huge expression when $n$ is large. Finally the fact that determinants exist turns out to be directly related to the fact that signs can be associated to all permutations in such a way that interchanging entries always changes the sign, which is part of the answer to question (3). As for question (1), we can now say that the determinant is uniquely determined by being an $n$-linear alternating expression in the entires of $n$ column vectors, which contains a term consisting of the product of their coordinates 1,2,...,$n$ in that order (the diagonal term) with coefficient $+1$. The explicit expression is a sum over all $n!$ permutations, the corresponding term being obtained by applying those coordinates in permuted order, and with the sign of the permutation as coefficient. A lot more can be said about question (2), but I'll stop here. |
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I recorded a lecture on the geometric definition of determinants: Geometric definition of determinants It has elements from the answers by Katie Banks and John Cook, and goes into details in a leisurely manner. |
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If you have a matrix
If you do a eigenvalue decomposition of $G$ you get eigenvalues $\lambda$ and eigenvectors $v$, that in combination $\lambda\times v$ describes the same space. Now there is the following equation, saying:
I.e., if you have a $3\times3$ matrix $H$ then $G$ is $3\times3$ too giving us three eigenvalues. The product of these eigenvalues give as the volume of a cuboid. With every extra dimension/eigenvalue the cuboid gets an extra dimension. |
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Think about a scalar equation, $$ax = b$$ where we want to solve for $x$. We know we can always solve the equation if $a\neq 0$, however, if $a=0$ then the answer is "it depends". If $b\neq 0$, then we cannot solve it, however, if $b=0$ then there are many solutions (i.e. $x \in \mathbb{R}$). The key point is that the ability to solve the equation unambiguously depends on whether $a=0$. When we consider the similar equation for matrices $$\mathbf{Ax} = \mathbf{b}$$ the question as to whether we can solve it is not so easily settled by whether $\mathbf{A}=\mathbf{0}$ because $\mathbf{A}$ could consist of all non-zero elements and still not be solvable for $\mathbf{b}\neq\mathbf{0}$. In fact, for two different vectors $\mathbf{y}_1 \neq \mathbf{0}$ and $\mathbf{y}_2\neq \mathbf{0}$ we could very well have that $$\mathbf{Ay}_1 \neq \mathbf{0}$$ and $$\mathbf{Ay}_2 = \mathbf{0}.$$ If we think of $\mathbf{y}$ as a vector, then there are some directions in which $\mathbf{A}$ behaves like non-zero (this is called the row space) and other directions where $\mathbf{A}$ behaves like zero (this is called the null space). The bottom line is that if $\mathbf{A}$ behaves like zero in some directions, then the answer to the question "is $\mathbf{Ax} = \mathbf{b}$ generally solvable for any $\mathbf{b}$?" is "it depends on $\mathbf{b}$". More specifically, if $\mathbf{b}$ is in the column space of $\mathbf{A}$, then there is a solution. So is there a way that we can tell whether $\mathbf{A}$ behaves like zero in some directions? Yes, it is the determinant! If $\det(\mathbf{A})\neq 0$ then $\mathbf{Ax} = \mathbf{b}$ always has a solution. However if, $\det(\mathbf{A}) = 0$ then $\mathbf{Ax} = \mathbf{b}$ may or may not have a solution depending on $\mathbf{b}$ and if there is one, then there are an infinite number of solutions. |
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now, to get an intuitive idea we first have to consider a set of simultaneous linear equqtions .In order to get an answer we should eliminate one variable and equate another.......... for example 2a+3b=13 and a+b=5 equating this we get a=2 and b=3 ........... but the thing toobserve is if the equations are proportional then it is impossible to come to an conclusion...... like.... a+b=2 and 2a+2b=4. we cannot find an solution to this bcoz theyre nothing but the same equations!!!(hint: divide eq. 2a+2b=4 by 2) ..... and hence determinant are used to determine the consistency of an equation here the first set is consistent(can be verified easily) .... but the second one is not bcoz the two rows are proportional!! |
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