Find Hessian Matrix Calculator

Calculate the Hessian Matrix

Enter the second partial derivatives (f_xx, f_xy, f_yy) as expressions of x and y, and the point (x₀, y₀) at which to evaluate them.

What is a Hessian Matrix Calculator?

A Hessian Matrix Calculator is a tool used to compute the Hessian matrix of a function of two or more variables at a specific point. The Hessian matrix is a square matrix of second-order partial derivatives of a scalar-valued function, or scalar field. It describes the local curvature of a function of many variables.

For a function f(x, y) of two variables, the Hessian matrix H is given by:

H =

fxx	fxy
fyx	fyy

Where f_xx is the second partial derivative with respect to x, f_xy is the mixed partial derivative (first with respect to x, then y), and so on. If the second partial derivatives are continuous, then f_xy = f_yx (Clairaut's theorem), and the Hessian matrix is symmetric.

This Hessian Matrix Calculator specifically helps you find the values of these second partial derivatives at a point (x₀, y₀) and construct the matrix, as well as calculate its determinant.

Who should use it?

Students of multivariable calculus, engineers, physicists, economists, and data scientists often use the Hessian matrix and might find a Hessian Matrix Calculator useful. It's crucial for optimization problems (finding local maxima, minima, or saddle points) and in understanding the local behavior of functions.

Common misconceptions

A common misconception is that the Hessian matrix directly gives the maximum or minimum. Instead, the Hessian matrix and its determinant, evaluated at a critical point (where the first derivatives are zero), help classify the critical point using the second derivative test for multivariable functions. A Hessian Matrix Calculator provides the data for this test.

Hessian Matrix Formula and Mathematical Explanation

For a function f(x, y), the Hessian matrix H(f) or ∇²f at a point (x₀, y₀) is defined as:

H(x₀, y₀) =

fxx(x0, y0)	fxy(x0, y0)
fyx(x0, y0)	fyy(x0, y0)

The components are:

• f_xx = ∂²f/∂x² (the second partial derivative with respect to x)
• f_xy = ∂²f/∂y∂x (the mixed partial derivative, first x then y)
• f_yx = ∂²f/∂x∂y (the mixed partial derivative, first y then x)
• f_yy = ∂²f/∂y² (the second partial derivative with respect to y)

If f has continuous second partial derivatives, then f_xy = f_yx.

The determinant of the Hessian matrix, D(x₀, y₀), is calculated as:

D = f_xx(x₀, y₀) * f_yy(x₀, y₀) – [f_xy(x₀, y₀)]²

This determinant, along with the value of f_xx(x₀, y₀), is used in the second derivative test at critical points (where ∇f = 0):

• If D > 0 and f_xx > 0, f has a local minimum at (x₀, y₀).
• If D > 0 and f_xx < 0, f has a local maximum at (x₀, y₀).
• If D < 0, f has a saddle point at (x₀, y₀).
• If D = 0, the test is inconclusive.

Variables Table

Variable	Meaning	Unit	Typical Range
f(x, y)	The function of two variables	Depends on f	Any real-valued function
x0, y0	The point at which the Hessian is evaluated	Depends on x, y	Real numbers
fxx, fxy, fyy	Second partial derivatives of f	Depends on f	Real numbers or expressions
H	Hessian Matrix	Matrix	2×2 Matrix of real numbers
D	Determinant of the Hessian Matrix	Depends on f	Real number

Variables involved in the Hessian Matrix calculation.

Practical Examples (Real-World Use Cases)

Example 1: Finding Local Extrema

Consider the function f(x, y) = x³ + y³ – 3xy. We want to analyze the critical point (1, 1). First, we find the first and second partial derivatives:

• f_x = 3x² – 3y, f_y = 3y² – 3x
• f_xx = 6x, f_yy = 6y, f_xy = -3

Using the Hessian Matrix Calculator with f_xx = 6x, f_xy = -3, f_yy = 6y, and the point (x₀, y₀) = (1, 1):

• f_xx(1, 1) = 6(1) = 6
• f_xy(1, 1) = -3
• f_yy(1, 1) = 6(1) = 6

Hessian matrix at (1, 1) is [[6, -3], [-3, 6]].

Determinant D = (6)(6) – (-3)² = 36 – 9 = 27.

Since D = 27 > 0 and f_xx = 6 > 0, the function has a local minimum at (1, 1).

Example 2: Identifying a Saddle Point

Consider the function g(x, y) = x² – y². The only critical point is (0, 0).

• g_x = 2x, g_y = -2y
• g_xx = 2, g_yy = -2, g_xy = 0

Using the Hessian Matrix Calculator with f_xx = 2, f_xy = 0, f_yy = -2, and (x₀, y₀) = (0, 0):

• f_xx(0, 0) = 2
• f_xy(0, 0) = 0
• f_yy(0, 0) = -2

Hessian matrix at (0, 0) is [[2, 0], [0, -2]].

Determinant D = (2)(-2) – (0)² = -4.

Since D = -4 < 0, the function has a saddle point at (0, 0).

How to Use This Hessian Matrix Calculator

1. Enter Second Partial Derivatives: Input the expressions for f_xx, f_xy, and f_yy in terms of x and y into the respective fields. For example, if f(x, y) = x²y + y³, then f_xx=2y, f_xy=2x, f_yy=6y.
2. Enter the Point (x₀, y₀): Input the x-coordinate (x₀) and y-coordinate (y₀) of the point where you want to evaluate the Hessian matrix.
3. Calculate: The calculator automatically updates the results as you type. You can also click "Calculate".
4. Read Results: The calculator displays the values of f_xx, f_xy, f_yy at (x₀, y₀), the Hessian matrix H, and its determinant D. The primary result indicates the nature of the point if it were a critical point (based on D and f_xx).
5. Interpret: If (x₀, y₀) is a critical point of f (i.e., f_x(x₀, y₀)=0 and f_y(x₀, y₀)=0), use the values of D and f_xx to determine if it's a local minimum, maximum, or saddle point.
6. Reset: Click "Reset" to clear inputs to default values.
7. Copy Results: Click "Copy Results" to copy the main findings to your clipboard.

This Hessian Matrix Calculator simplifies the process of evaluating the Hessian and its determinant at a specific point, especially when the derivatives are complex.

Key Factors That Affect Hessian Matrix Results

The Hessian matrix and its determinant at a point (x₀, y₀) are entirely determined by:

1. The Function f(x, y): The form of the function f dictates its second partial derivatives f_xx, f_xy, and f_yy. Different functions will have different Hessian matrices.
2. The Point (x₀, y₀): The specific values of x₀ and y₀ at which the second partial derivatives are evaluated determine the numerical values in the Hessian matrix and its determinant. The nature of the function's curvature can change from point to point.
3. Continuity of Second Partials: The assumption f_xy = f_yx relies on the continuity of these mixed partial derivatives. If they are not continuous, the Hessian might not be symmetric. Our Hessian Matrix Calculator assumes symmetry.
4. Whether (x₀, y₀) is a Critical Point: The interpretation of the Hessian matrix in terms of local extrema (min/max/saddle) is most meaningful at critical points where the first derivatives (gradient) are zero.
5. The Coordinate System: While not directly an input to this calculator, the form of the function and its derivatives depends on the coordinate system used (e.g., Cartesian, polar).
6. Higher-Order Terms: If the determinant D=0, the second derivative test is inconclusive, and higher-order derivatives might be needed to classify the critical point, which are not part of the standard Hessian matrix.

Frequently Asked Questions (FAQ)

What does the determinant of the Hessian matrix tell us?

At a critical point, the sign of the determinant (D), combined with the sign of f_xx, helps classify the point as a local minimum (D>0, f_xx>0), local maximum (D>0, f_xx<0), or saddle point (D<0). If D=0, the test is inconclusive.

Can I use this Hessian Matrix Calculator for functions of one variable?

The Hessian matrix is defined for functions of two or more variables. For a function of one variable, f(x), you'd look at the second derivative f"(x) to determine concavity and local extrema.

What if my function has more than two variables?

This specific Hessian Matrix Calculator is designed for functions of two variables (x, y), resulting in a 2×2 Hessian. For functions of n variables, the Hessian is an nxn matrix, and you'd need a more general tool or manual calculation.

What if f_xy is not equal to f_yx?

Clairaut's theorem states that if the second partial derivatives are continuous, then f_xy = f_yx. Most functions encountered in basic calculus and applications satisfy this. If they are not equal, the Hessian matrix is not symmetric, but this calculator assumes f_xy=f_yx based on the f_xy input.

Why does the calculator ask for f_xx, f_xy, f_yy as expressions?

To avoid the complexity of symbolic differentiation from f(x,y) directly within the browser without external libraries, the calculator requires you to input the second partial derivatives as expressions in x and y, which it then evaluates at (x₀, y₀).

What happens if D=0?

If the determinant D=0 at a critical point, the second derivative test is inconclusive. The point could be a local extremum, a saddle point, or something else. Higher-order derivative tests might be needed.

How do I find the critical points to test with the Hessian?

To find critical points, you need to solve the system of equations f_x(x, y) = 0 and f_y(x, y) = 0 simultaneously. The solutions (x₀, y₀) are the critical points.

Is the Hessian matrix always symmetric?

It is symmetric if the mixed second partial derivatives (f_xy and f_yx) are continuous. In most practical cases, yes.

Related Tools and Internal Resources

• Gradient Calculator: Find the gradient (vector of first partial derivatives) of a multivariable function.
• Partial Derivative Calculator: Calculate partial derivatives of functions with respect to different variables.
• Matrix Determinant Calculator: Calculate the determinant of a given matrix.
• Critical Point Finder: Helps find points where the gradient of a function is zero.
• Local Extrema Calculator: Uses first and second derivative tests to find local minima and maxima.
• Saddle Point Calculator: Identifies saddle points of a function of two variables.